Magma Chambers: Formation, Local Stresses, Excess Pressures, and Compartments

Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41

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Journal of Volcanology and Geothermal Research

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Review Magma chambers: Formation, local stresses, excess pressures, and compartments

Agust Gudmundsson

Department of Earth Sciences, Royal Holloway University of London, Egham TW20 0EX, UK article info abstract

Article history: An existing magma chamber is normally a necessary condition for the generation of a large volcanic edifice. Received 7 January 2012 Most magma chambers form through repeated magma injections, commonly sills, and gradually expand and Accepted 20 May 2012 change their shapes. Highly irregular magma-chamber shapes are thermo-mechanically unstable; common Available online 1 June 2012 long-term equilibrium shapes are comparatively smooth and approximate those of ellipsoids of revolution. Some chambers, particularly small and sill-like, may be totally molten. Most chambers, however, are only Keywords: partially molten, the main part of the chamber being crystal mush, a porous material. During an eruption, Magma chambers Crustal stresses magma is drawn from the crystal mush towards a molten zone beneath the lower end of the feeder dyke. Stress concentrations Magma transport to the feeder dyke, however, depends on the chamber's internal structure; in particular Dykes on whether the chamber contains pressure compartments that are, to a degree, isolated from other compart- Sills ments. It is only during large drops in the hydraulic potential beneath the feeder dyke that other compart- Volcanic eruptions ments become likely to supply magma to the erupting compartment, thereby contributing to its excess pressure (the pressure needed to rupture a magma chamber) and the duration of the eruption. Simple analytical models suggest that during a typical eruption, the excess-pressure in the chamber decreases exponentially. This result applies to a magma chamber that (a) is homogeneous and totally fluid (contains no compartments), (b) is not subject to significant replenishment (inflow of new magma into the chamber) during the eruption, and (c) contains magma where exsolution of gas has no significant effect on the excess pressure. For a chamber consisting of pressure compartments, the exponential excess-pressure decline applies primarily to a single erupting compartment. When more than one compartment contributes magma to the eruption, the excess pressure may decline much more slowly and irregularly. Excess pressure is normally similar to the in-situ tensile strength of the host rock, 0.5–9 MPa. These in- situ strength estimates are based on hydraulic fracture measurements in drill-holes worldwide down to crustal depths of about 9 km. These measurements do not support some recent magma-chamber stress models that predict (a) extra gravity-related wall-parallel stresses at the boundaries of magma chambers and (b) magma-chamber excess pressures prior to rupture of as much as hundreds of mega-pascals, particularly at great depths. General stress models of magma chambers are of two main types: analytical and numerical. Earlier analytical models were based on a nucleus-of-strain source (a ‘point pressure source’) for the magma chamber, and have been very useful for rough estimates of magma-chamber depths from surface deformation during unrest periods. More recent models assume the magma chamber to be axisymmetric ellipsoids or, in two-dimensions, ellipses of various shapes. Nearly all these models use the excess pressure in the chamber as the only loading (since lithostatic stress effects are then automatically taken into account), assume the chamber to be totally molten, and predict similar local stress fields. The predicted stress fields are generally in agreement with the world-wide stress measurements in drill-holes and, in particular, with the in-situ tensile-strength estimates. Recent numerical models consider magma-chambers of various (ideal) shapes and sizes in relation to their depths below the Earth's surface. They also take into account crustal heterogeneities and anisotropies; in particular the effects of the effects of a nearby free surface and horizontal and inclined (dipping) mechanical layering. The results show that the free surface may have strong effects on the local stresses if the chamber is comparatively close to the surface. The mechanical layering, however, may have even stronger effects. For realistic layering, and other heterogeneities, the numerical models predict complex local stresses around magma chambers, with implications for dyke paths, dyke arrest, and ring-fault formation. © 2012 Elsevier B.V. All rights reserved.

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Contents

1. Introduction ...... 20 2. Deﬁnition of a magma chamber ...... 21 3. Formation and geometry of magma chambers ...... 21 4. Crustal stresses ...... 27 5. Stresses and pressures associated with dykes and sills ...... 28 6. Stresses around magma chambers ...... 32 6.1. Stresses at magma-chamber initiation ...... 32 6.2. Different approach to magma-chamber stress modelling ...... 33 6.2.1. The ﬁrst point ...... 33 6.2.2. The second point ...... 34 6.2.3. The third point ...... 34 7. Chamber pressure variation during an eruption: effects of compartments ...... 34 8. Discussion ...... 37 9. Conclusions ...... 38 Acknowledgements ...... 39 References ...... 39

1. Introduction as regards volume and particularly as regards frequency, along active volcanic zones and fields. A magma chamber is the heart of every active major polygenetic In order to understand volcano behaviour, in particular the size- volcano (Fig. 1). Thus, each major volcanic structure, such as a strato- distributions, the volumetric flow rates, and the duration of its erup- volcano, a collapse caldera, or a large shield volcano (a basaltic edi- tions, it is necessary to have a reliable knowledge of the mechanical fice), is supplied with magma from a (comparatively shallow) behaviour of the associated magma chamber. That knowledge should crustal magma chamber. Formation of a major volcanic edifice is the include the following points: consequence of the existence of a magma chamber—not the other way around. The chamber acts as a collector of magma from the (1) How the host rock and the matrix of the chamber itself re- deeper source (here referred to as a reservoir) and channels that spond to stress and fluid pressure changes; magma to a limited area at the surface above where the volcano (2) What the material properties of the chamber and its host rock builds up. If there were no shallow magma chambers, the volcanic ac- are and how these may change with time; tivity would be much more evenly distributed than it actually is, both (3) How the shape and loading of the chamber affects the local stress fields inside and outside the chamber; (4) The likelihood of rupture with a dyke, sheet, or sill injection and, eventually, eruption.

For an eruption to occur, the necessary condition is that the magma chamber or reservoir ruptures and a fluid-driven fracture (usually a dyke or an inclined sheet) is able to propagate from the chamber to the surface. In some volcanoes, such as Stromboli (Italy) and Sakura–Jima (Japan), as well as in some active lava lakes, such as in Kilauea (Hawaii), Nyiragongo and Erta Ale (Africa), and Mount Erebus (Antarctica), there is quasi-continuous eruptive activity over long periods of time (Simkin and Siebert, 2000; Frank, 2003; Rosi et al., 2003; Siebert et al., 2010). In these cases, the conduit may be con- tinuously open. However, volcanoes with these types of eruptions are rare in comparison with those where a new fluid-driven fracture forms during each eruption. Also, it is likely that even if some of the quasi-continuous eruptions may last for tens, hundreds or (in case of Stromboli) perhaps a couple of thousand years (Kilburn and McGuire, 2001; Rosi et al., 2003; Siebert et al., 2010), the continuous eruptions are short-lived in comparison with the overall lifetime of the volcano itself—of the order of 105 to 106 years. Thus, for most of the time, in most volcanoes, and for most eruptions, a magma- chamber rupture and fluid-driven fracture propagation to the surface is the mechanism of eruption. Both the magma-chamber rupture and the fluid-driven fracture propagation to the surface depend primarily on the local stresses inside the volcano and around the magma chamber. This has been widely recognised in the past decades, with many papers and books Fig. 1. Schematic illustration of a volcanic edifice, here a stratovolcano. The magma focusing on modelling the stress and displacement fields in volca- chamber acts as a sink for magma from a deeper magma accumulation zone (here re- noes. These models fall broadly into two basic groups. First, those ferred to as a reservoir) and a source for inclined sheets, sills, and dykes (including that attempt primarily to explain the surface deformation and the feeder dykes). Many of the dykes become arrested, some becoming deflected into sills. The volcano builds up into a cone because the chamber channels magma to a lim- depth to the associated magma chamber during unrest periods. ited area on the surface. Second, those that attempt to model the stress fields around the A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41 21 magma chamber and inside the volcano itself in relation to the condi- and the duration, of the eruption. The third aim of the paper is to tions for rupture and dyke/sheet propagation to the surface. provide simple analytical results on excess-pressure variation in a In the first group, where the main aim is to understand surface de- chamber during eruption. In detail, the excess-pressure (and compo- formation during unrest periods and eruptions, the best known is the sitional) variation during an eruption depend on whether the cham- ‘Mogi model’ (Mogi, 1958). This model, initially derived from nucleus- ber contains pressure compartments, as are well known from many of-strain solutions in solid mechanics (Melan, 1932; Mindlin, 1936) hydrocarbon reservoirs. Compartments depend on the local stresses, and applied to volcanoes by Anderson (1936), focuses on explaining pressures, and mechanical properties of the chamber. The fourth volcano surface deformation during an unrest period in terms of the and final aim is thus to touch briefly on the topic of magma- depth and pressure of a magma chamber (modelled as a nucleus of chamber compartments. The paper provides a review and analysis strain, that is, a point pressure source). Since the magma chamber is of many existing ideas and models, but presents also some new regarded as the product of the source radius and the excess pressure, ideas, particularly as regards excess-pressure variations and compart- the excess pressure and the size of the chamber cannot be estimated in- ments in magma chambers. dependently. There is thus no information provided as to the stresses that control magma-chamber rupture, namely the stresses that concen- 2. Definition of a magma chamber trate around the magma chamber itself. The ‘Mogi model’ and its application to surface deformation, including caldera formation, is discussed A magma chamber is a partially or totally molten body located in the in detail by Mogi (1958), Dzurisin (2006), Kusumoto and Takemura crust and supplied with magma from a deeper source, a reservoir (2005), Gudmundsson (2006), Sturkell et al. (2006), Poland et al. (Fig. 1). While active, a magma chamber acts as a sink for magma from (2006), Kusumoto and Gudmundsson (2009),andSegall (2010). the deeper reservoir, and as a source for magma injections (dykes, Extensions of the ‘Mogi model’ include those of Davis (1986) sheets, sills) into the surrounding crust and the associated volcano. where the chamber is no longer assumed spherical (a point source) Some magma chambers, particularly small sill-like chambers during but rather an arbitrary oriented triaxial ellipsoidal cavity. As a rule, their early stages of development, may be totally molten. Other cham- however, the nucleus-of-strain models regard the crustal segment bers, however, are partially molten from their initiation, or become so hosting the magma chamber as a homogeneous, isotropic, elastic quickly during their evolution. A large part of the chamber may then half-space, thereby ignoring all mechanical layering and heterogene- consist of a crystal mush, a matrix, which behaves as poroelastic ities of the segment. Masterlark (2007) found significant differences (Maaloe and Scheie, 1982; McKenzie, 1984; Gudmundsson, 1987; between depth estimates for magma chambers based on homoge- Marsh, 1989, 2000; Sinton and Detrick, 1992). Under these conditions, neous and isotropic elastic half spaces and those based on layered only the central or upper part of the chamber may be totally molten crustal segments. Also, numerical models indicate that the surface and the lower, and greater, part a hot, partially molten crystalline stress and deformation above magma chambers and dykes, for exam- mush or matrix. This latter appears to be common for the magma cham- ple, depend strongly on the mechanical layering of the crustal seg- bers at mid-ocean ridges (Macdonald, 1982; Sinton and Detrick, 1992; ments hosting these structures (Gudmundsson, 2011a). Using Mutter et al., 1995; MacLeod and Yaouancq, 2000; Singh et al., 2006; numerical and analytical methods, recent models consider the effects Canales et al., 2009). of heterogeneous, layered, and anelastic crustal segments to infer An active (fluid) magma chamber has clearly mechanical proper- magma-chamber shape and inflation mechanisms (e.g., Bonafede et ties that differ from those of the host rock. But even a solidified al., 1986; Bonaccorso et al., 2005; Trasatti et al., 2005; Bonafede and magma chamber, a pluton, with the same chemical composition as Ferrari, 2009; Trasatti et al., 2011). the host rock (which is rare) may differ mechanically from the host In the second group, the main aim of modelling is to understand rock. This applies particularly when the pluton has a higher tempera- the stress (and displacement) fields inside the volcano and how ture than the host rock and follows because the mechanical proper- they affect magma-chamber rupture, dyke/sheet propagation, and ties of rocks change with temperature. In particular, Young's the likelihood of eruption during an unrest period. These models are modulus (stiffness) depends on temperature (Balme et al., 2004). also used to explain the surface deformation, not only in terms of When the chamber is totally molten, it acts as a fluid-filled cavity, magma-chamber inflation and deflation, but also as regards the stress and when it is partially molten or, alternatively, solidified but still and deformation induced at the surface by upward-propagating hot, as a comparatively compliant (soft) inclusion (Andrew and dykes. This second group of models has received increasing attention Gudmundsson, 2008; Gudmundsson, 2011a). When the solidified in the past few decades. However, while there is general agreement chamber rock has reached the same temperature as the host rock, as to some aspects of the modelling, such as the common magma- the chamber is referred to as a pluton. Most plutons are of rock chamber shapes, there are divergent views as to the theoretical types that differ mechanically from those of the host rocks: for exam- stresses around magma chambers and how to model them. ple, the pluton may be a gabbro body located in a (mostly) basaltic As suggested in the invitation to write this paper, one of its princi- lava pile (Fig. 2). The pluton (the fossil magma chamber) then acts pal aims is to discuss these divergent views and to clarify how the as a stiff inclusion. All cavities and inclusions modify the local stress state of stress around a magma chamber should be calculated and field and concentrate stresses. However, the induced stresses depend compared with in-situ measurements of stresses and strengths in much on the geometry of the cavity/inclusion (Savin, 1961; Boresi drill-holes worldwide. Since the stress field inside a volcano, and in and Sidebottom, 1985; Tan, 1994; Saada, 2009). This is one reason particular around its magma chamber, largely controls the volcano why knowing the shape of a magma chamber, active or fossil, is of deformation and the likelihood and duration of its eruptions, it is of great importance for understanding the local stress field around the fundamental importance that there should be a clear theory as to chamber. how to calculate the local stress field around a magma chamber. The local stress field around a chamber and inside the associated vol- 3. Formation and geometry of magma chambers cano, however, depend much on the shape and size of the chamber which, in turn, are related to the initiation and evolution of the cham- Most crustal magma chambers presumably develop from sills ber. A second aim of this paper is thus to review various likely scenar- (Pollard and Johnson, 1973; Gudmundsson, 1990; Annen and Sparks, ios for the initiation and geometric development of magma chambers. 2002; Kavanagh et al., 2006; Menand, 2008; Menand et al., 2010; The local stresses inside a volcano determine not only the condition Menand, 2011; Menand et al., 2011). In fact, many chambers, particu- for magma-chamber rupture and dyke propagation to the surface, larly at mid-ocean ridges, maintain the sill-like geometry of at least but also affect the behaviour, in particular the volumetric flow rate the molten part throughout their lifetimes (Macdonald, 1982; Sinton 22 A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41

Fig. 2. Part of a fossil magma chamber in Geitafell, Southeast Iceland. View north, the exposed part of the chamber, a gabbro pluton, is at a depth of about 2 km below the initial top of the associated volcano and is located within a pile of (mostly) basaltic lava ﬂows with mechanical properties that differ from those of the pluton. At the east (right) contact (marked) of the pluton, there is an abrupt change from gabbro to close to 100% inclined sheets and dykes (cf. Gudmundsson, 2011b, Fig. 3). The width (lateral dimension) of the part of the gabbro pluton exposed here is about 400 m and the height of the cliffs as see here is about 80 m. and Detrick, 1992; Mutter et al., 1995; MacLeod and Yaouancq, 2000; (Fig. 3a,c), concave-upwards bending (Fig. 3b), and stair-case Singh et al., 2006; Canales et al., 2009). Most sills, however, do not de- (Fig. 3d), of which the saucer-shape is a special case. Many magma velop into magma chambers. Thus, special conditions must be met for chambers that have been detected beneath mid-ocean ridges appear a sill to evolve into a magma chamber. These include that the initial comparatively straight, as inferred from seismic studies. Such seismic sill must (1) be comparatively thick, normally at least tens of metres studies, however, have generally a low resolution and do not allow us (but depends on the spreading rate) and (2) receive magma so fre- to decide on the detailed shapes. quently (through dykes) that it stays liquid for a considerable time Calculations show that the chances of a sill developing into a and has the chance to grow into a chamber (Gudmundsson, 1990; magma chamber in an area undergoing extension (such as at a diver- Menand et al., 2010, 2011). The formation of sills, that is, the conditions gent plate boundary) depend largely on a combination of spreading for dykes deﬂecting into sills, is discussed by Pollard and Johnson rates and cooling rates (Gudmundsson, 1990). This follows because (1973), Menand (2008), Gudmundsson (2011b), Menand et al. (2010, the rate of injection of dykes that could meet the sill and supply 2011),andMaccaferri et al. (2011). magma to it is directly proportional to the associated spreading If most magma chambers initiate from sills, then it follows that the rate. The higher the dyke-injection rate, the greater the chances of a initial chamber shape is sill-like. Sills, however, have many different dyke meeting the sill, and thus the greater the probability of that shapes (Fig. 3). Perhaps the most common sill shapes are straight sill developing into a magma chamber. This conclusion is supported

Fig. 3. Various stress and mechanical conditions favour the deflection of dykes into sills (cf. Kavanagh et al., 2006; Gudmundsson, 2011b; Menand, 2011; Menand et al., 2011). The resulting sills can have various forms, some of which are shown here. (a) Double-deflected straight sill. (b) Concave or upward-bending sill. (b) Single-deflected straight sill. (d) Stair-case-shaped double-deflected sill, a subgroup of which is the saucer-shaped (disc-shaped) sill. (e) A single-deflected sill arrested by a fault. (f) A single-deflected sill that propagates towards and up along the fault; the sill follows the fault for a while and then propagates parallel with a contact in the footwall. A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41 23 by there being many more, and more extensive, shallow magma be identified. This is primarily because of the more highly viscous chambers at fast-spreading ridges than at slow-spreading ridges magma of the sills makes mixing of the new and old magma, if both (Macdonald, 1982; Sinton and Detrick, 1992; Mutter et al., 1995; are acid, less likely than in the case of basaltic sills. One particularly MacLeod and Yaouancq, 2000; Singh et al., 2006; Canales et al., 2009). clear example of a fossil acid magma chamber formed by sill injections In mafic (basic) fossil magma chambers, the contacts between in- occurs in Slaufrudalur in East Iceland (Fig. 5; Gudmundsson, 2011b). dividual sills injected during the formation of the magma chamber Here the contacts between the sills are clearly seen since they form dis- cannot normally be identified. This is because the low-viscosity basal- tinct layers. The layers dip 5–10°NW, similar to that of the lava flows tic magma supplied through a new sill injection normally becomes that form the host rock of the pluton, and their thicknesses are mostly mixed with the existing magma in the earlier sills. When multiple ba- 15–50 m (Fig. 5; Beswick, 1965). This magma chamber formed by the saltic sills are seen and clearly identified as such in the field (Fig. 4), it piling up of 15–50 m thick acid (granophyric) sills. With an exposed is because they formed over a time period too long for the successive maximum thickness of about 700 m, it clearly formed through many sill injections to develop into a single magma body, a chamber. sill injections. There are many other examples of felsic magma cham- By contrast, in some felsic (acid) fossil magma chambers formed by ber/pluton formation through numerous, commonly sill-like, injections. sill injections, and presently exposed as plutons, the individual sills can For example, many large granitic plutons are now thought to have

Fig. 4. (a) Part of a thick basaltic sill in East Iceland. The sill is 120-m thick, composed of at least 16 columnar rows (some indicated) and thus a multiple intrusion. The overall shape of the sill is concave (Fig. 3b), as is seen from a greater distance (Gudmundsson, 2011b, Fig. 12), and is located at a depth of about 800 m below the top of the rift zone at the time of sill emplacement. The exposed lateral dimension of the sill is about 3 km, but the rest is eroded away so that its initial dimension is unknown. The sill did not develop into a magma chamber, presumably because the rate of dyke injections was so low that individual columnar rows became solidiﬁed before the subsequent ones were injected (cf. Fig. 8). (b) View east, a cluster or complex of sills in the caldera walls of Las Canadas in Tenerife (Canary Islands). This cluster formed at about 100 m below the free surface of the volcano; a crater is seen at the surface (and its feeder dyke is indicated; see Fig. 13 for the details of the feeder). The sill cluster (marked by Sill 1, Sill 2, Sill 3 and Sill 4) extends laterally for many hundred metres (only a part of the complex is seen here) and reaches thicknesses of tens of metres. Here, however, the total thickness of the cluster is about 20 m. The sills formed at contacts between mechanically dissimilar pyroclastic rocks (light-brown to dark-brown in colour; one layer indicated) and elsewhere in the wall at contacts between stiff lava ﬂows and pyroclastic rocks. 24 A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41

Fig. 5. Part of the fossil magma chamber at Slaufrudalur, Southeast Iceland (located in Gudmundsson, 2011b, Figs. 4, 14). (a) View northwest, the present pluton is composed of granophyre and has an exposed volume of about 10 km3. Many of the walls are well exposed—two are indicated here. Part of the roof is also exposed, through which many felsic dykes have been injected. Many of these dykes presumably became arrested, that is, were non-feeders, which is one major reason why their contacts with the magma chamber remained filled with magma (did not close, as they normally would tend to do at the end of an associated eruption). The vertical stress at points A and B refers to the discussion about ‘wall-parallel’ stresses in Section 6. Fig. 5b. (b) The Slaufrudalur magma chamber was generated through injection of numerous granophyre sills. Some of the sills can be seen as crude layering, most of the layers (sills) being 15–50 m thick (Beswick, 1965; cf. Gudmundsson, 2011b). formed through many dyke-fed injections of sills or sheets, generating means that the space for the intrusion is generated by the fluid- an overall flat-lying (sill like) magma body (Petford et al., 2000). overpressure driven opening displacement of the fracture walls. Lacco- The mode of emplacement of the Slaufrudalur sills, as for most liths, in particular, show clear evidence of bending and fracturing of the sills, was forceful, that is, the host rock became deformed. This is eas- roof layers (Fig. 6; Hawkes and Hawkes, 1933; Pollard and Johnson, ily seen because of change in the dip of the host rock close to the plu- 1973; Pasquare and Tibaldi, 2007). The roofs of sills emplaced in sedi- ton. In the basaltic lava pile west of Slaufrudalur (not seen in Fig. 5) mentary rocks also commonly show evidence of forceful emplacement the lavas are not disturbed by the pluton and the regional dip is (Hansen and Cartwright, 2006). A large fraction of the space needed for 8–10°NW, and thus towards the pluton. But close to the pluton, at the sills and other magma-chamber intrusions is thus generated by the entrance of the valley of Slaufrudalur (Fig. 5) the dip of the lava forceful uplift or up-bending of the roof, as well as down-bending of flows is 11°E, that is, away from the pluton. It follows that the lava the floor. There are other factors that may significantly affect the exact pile has been tilted by 15–20° during the emplacement of the emplacement mechanics, in particular whether up-bending (as in Slaufrudalur magma chamber. The dykes and inclined sheets that dis- laccoliths) or down-bending (as presumably in many sill-like intru- sect the roof of the pluton (Fig. 5) demonstrate that it functioned as a sions) dominates (Petraske et al., 1978). These include the mechanical magma chamber, that is, sent off dykes. Presumably, some of the layering of the host rock, the depth of emplacement, and the size of injected dykes supplied magma to a volcano at the surface. the sill-like intrusion. Most sills and laccoliths and other potential magma-chamber intru- The geometric evolution of the magma chamber after the initial sions show clear evidence of being forceful intrusions (Fig. 6). This sill emplacement varies considerably and depends on many factors A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41 25

Fig. 6. Sandfell, a felsic laccolith in East Iceland. Like most plutons in Iceland, Sandfell shows clear evidence of being a forceful intrusion; in this case, up-doming or tilting of the lava flows close to and on the top of the laccolith. The local dips of the lava flows reach 34–36°, whereas the regional dip of the lava flows (the ‘normal-dipping lava flows’), away from the laccolith, is about 8°. The top of the mountain is at 743 m a.s.l., and the thickness of the laccolith itself is thought to be around 500 m (cf. Hawkes and Hawkes, 1933).

(Fig. 7). Perhaps the most common type of magma chamber, howev- common than sills or larger plutons, so that we may assume that, er, is sill-like (Figs. 7d, 8). This is the shape that is often imaged in even if magma chambers of such a geometry exist, they are not very seismic studies of volcanoes and rift zones, such as at mid-ocean common. Much more common are general oblate-ellipsoidal cham- ridges (Macdonald, 1982; Sinton and Detrick, 1992; Mutter et al., bers (Fig. 7d). These are similar to sill-like chambers but, partly be- 1995; MacLeod and Yaouancq, 2000; Singh et al., 2006; Canales et cause of the size and density of the magma, the magma pressure al., 2009). This magma-chamber shape that is supported by the com- may bend the layers above and below and forms a comparatively mon occurrence of sills in volcanic areas and the formation of collapse thick, oblate-shaped chamber. Many large chambers worldwide are calderas. Generally, a sill-like magma chamber is the geometry most apparently of this shape (Petraske et al., 1978; Sibbett, 1988; Marsh, favourable for the generation of the ring-faults along which piston- 1989, 2000; Menand et al., 2011). like subsidence takes place (Geyer et al., 2006; Acocella, 2007; Bell-jar magma chambers have often been proposed for plutons Geyer and Marti, 2008, 2009; Gudmundsson, 2011a). and magma chambers (Anderson, 1936). A bell-jar intrusion is essen- When certain mechanical conditions are satisﬁed, a sill may devel- tially a ring fault that fails to reach the surface and whose vertical op into a laccolith (Pollard and Johnson, 1973; Pasquare and Tibaldi, parts connect through a subhorizontal fracture (for example, along 2007; Bunger and Cruden, 2011). Some magma chambers are clearly an existing contact between lava ﬂows). The idea is that magma is laccoliths (Fig. 6). However, in general, laccoliths are much less injected under the contact/roof and the host-rock below subsides

Fig. 7. Schematic illustration of the various possible shapes of magma chambers. (a) Chambers with very irregular boundaries (surfaces) are thermally and mechanically unstable and tend to smooth out the irregularities (cf. Fig. 9). (b) Roughly prolate ellipsoidal chambers may exist beneath some volcanic ediﬁces, particularly comparatively narrows cones with steep slopes. This follows because the magma-chamber rupture, and associated feeder dykes, would mostly be injected from the top of the chamber, indicating a narrow surface area for the eruptions. (c) Roughly spherical magma chambers may be common, particularly at the later stages of the chamber evolution. Many such chambers generate swarms of inclined sheets. (d) Roughly oblate ellipsoidal or sill-like chambers are presumably the most common chamber geometry. They are particularly common at mid- ocean ridges, but have been detected seismically beneath many volcanoes. 26 A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41

Fig. 8. Schematic illustration of a magma-chamber formation through the injection of sills. (a) A sill forms at the contact between mechanically dissimilar rock layers (cf. Menand, 2008, 2011; Gudmundsson, 2011b). (b) Subsequent dyke injections become arrested and their magmas partly absorbed by the original sill. (c) The sill cluster expands and (d) forms a sill-like magma chamber that supplies magma to a volcano. In this schematic illustration, the earlier sills are shown as cooling somewhat before a new sill emplacement occurs (sills number 1–3 in (c)). However, if the rate of injection is so low that the earlier sills solidify before the new injections occur, the cluster is unlikely to develop into a magma chamber and more likely to be recognisable as individual sills or a single sill with many columnar rows (Fig. 4). Here it is assumed that all the sills stay liquid during the formation of the chamber, even if they are shown as having cooled somewhat (the change in colour) between successive injections (c). This scenario is generally appropriate for both mafic and felsic sills, but in the latter case the sill contacts may still be seen even if the body acted as a single magma chamber (Fig. 5). into an existing magma chamber below, in a similar manner as conduits at shallow depths, perhaps at the intersections between piston-like subsidence occurs along ring faults. A classic example of major tectonic fractures. Fossil chambers of this shape are represent- a bell-jar intrusion is the Great Eucrite Intrusion, of gabbro, forming ed by plugs and necks, as are commonly seen at shallow depths in a part of the intrusive complex of Ardnamurchan in West Scotland eroded stratovolcanoes. (O'Driscoll et al., 2006). Spherical magma chambers are favoured in volcanoes subject to Bell-jar geometry has also been suggested for the Slaufrudalur Plu- an isotropic or close-to isotropic state of stress (Fig. 7c). But this ge- ton (Cargill et al., 1928). There is no doubt that the walls of ometry may also result directly from the cooling of a chamber that Slaufrudalur are faults (Fig. 5a), but there is no evidence that they originally had a different shape. When the rate of receiving new acted as ring dykes (Gudmundsson, 2011b). Furthermore, as indicat- magma does not keep up with the rate of cooling and solidification, ed above, there is clear evidence of forceful emplacement of the intru- the size of the fluid part of the magma chamber reduces. To minimise sion. The trend of the Slaufrudalur Pluton is parallel with the the rate of flow of heat out of the chamber, it may gradually evolve palaeorift zone, so that the faults are likely to be steeply dipping nor- into a spherical chamber (Fig. 7c). An original sill-like, oblate mal faults. Slip on graben faults is known to change the stress field so spheriodal or prolate spheroidal magma chamber may (Fig. 7), as to encourage dyke deflection into sills (Gudmundsson, 2011a). when the heat supply decreases, may change into a comparatively The evolution of the sills into a larger magma chamber is thus like- small, spherical magma chamber. Thus, many spherical chambers ly to be as follows (Fig. 8). A dyke becomes deflected into a sill at a may be remnants of earlier and larger chambers of different shapes. contact where the mechanical properties and changes in local stress- It is, however, also possible that some very long-lived large magma es favour sill formation (Gudmundsson, 1990, 2011b; Menand, 2008; chambers become close to spherical in shape, particularly if the Menand et al., 2011). If the dyke-injection frequency is so high that time-average regional state of stress is close to isotropic. This would the earlier sills do not have a chance of solidifying before new dykes be partly achieved through forceful up-bending and down-bending meet the sills, the dykes tend to be deflected again and again on of the host-rock layers above and below the chamber, and partly meeting the earlier sills (Fig. 8b–c; Gudmundsson, 2011b). Subse- through anatexis (melting) and perhaps some ductile deformation, quent sills are then emplaced (normally below) the earlier sills, of the host rock. thereby building up the magma chamber (Fig. 8d). A long-lived magma chamber cannot be very irregular in shape; it In case the dyke was injected into a graben, a recent slip on the tends to smooth out the initial surface irregularities during its lifetime graben faults would increase the horizontal compressive stress with- (Fig. 9). Thus, fossil magma chambers, plutons, tend to have compar- in the graben and thereby help trigger the dyke deflection into the atively smooth large-scale contacts with the surrounding rocks sill. If the supply of magma is high enough, the sill propagates lateral- (Figs. 5, 10a), even if some show evidence of complex interaction be- ly until it meets with one or both the normal fault (Fig. 3e). Many sills tween different magmas (Fig. 10b). That large-scale irregularities in in rift zones, such as in sedimentary basins, are known to terminate at contacts tend to smooth out follows from simple thermal consider- normal faults (Fjeldskaar et al., 2008). Thus, it may be that many of ations (Jaeger, 1961, 1964; Gudmundsson, 1990). For example, if an the intrusions classified as bell-jar are simply piling up of sills within irregular part of the chamber projects into the host rock, that part an active graben, as is likely to have been the case for formation of the has a comparatively large area of contact with the host rock, so that, Slaufrudalur magma chamber (Fig. 5). Depending on the stress situa- assuming that the host rock is cooler than the magma (as is normally tion, some sills are able to propagate upwards (or downwards) along the case), heat is transferred rapidly from the magma to the host rock. the fault for a while and then propagate laterally again (Fig. 3f). Most, This part of the chamber tends to cool down rapidly, become solidi- however, terminate at the faults (Fig. 3e). fied, and thus no longer a part of the active chamber. Conversely, if Some magma chambers eventually become spherical (Fig. 7c) or an irregular part of the host rock projects into the magma chamber, even prolate-ellipsoidal (Fig. 7b; Gudmundsson, 1988,1990). A heat is transferred rapidly from the magma into that part of the prolate-ellipsoidal chamber is most easily developed from cylindrical host rock which, thereby, tends to melt partially. Eventually, the A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41 27

force acts, the greater is the stress. In the simplest way, stress may be deﬁned as:

F σ ¼ : ð1Þ A

Here F is the force in newtons (N) and A is the area in square metres (m2) upon which the force acts. The symbol σ is used, as I do here, when the force that generates the stress is normal to the area A upon which it acts and is referred to as normal stress. By contrast, a shear force operates parallel to the plane of interest, such as a fault plane, and generates shear stress. Shear stress is here denoted by the symbol τ. The units of stress are Nm− 2, that is, newtons per square metre, referred to as pascals. A pascal, however, is a very small unit, roughly Fig. 9. Magma chamber with a very irregular boundary (surface) is mechanically and equal to the pressure given by water lens of thickness 1×10− 4 mor thermally unstable. It is mechanically unstable because the notches that project into 0.1 mm. Thus, in geology and many other sciences megapascals the host rock are stress raisers and would inject dykes if there was any non-zero excess pressure in the chamber. It is thermally unstable because the jogs that project into the (million pascals) are used for stress and pressure and strength, and chamber would tend to melt, and the notches that project into the host rock would gigapascals (thousand million pascals) for elastic moduli (such as tend to solidify. Gradually, the chamber would assume a more stable, smoother geom- Young's modulus and shear modulus) of the rock units and layers. etry, as indicated; commonly, similar to that in Fig. (18). Stress as defined in Eq. (1) is a vector, sometimes referred to as the traction, the traction vector, or simply the stress vector. It is a vector whole part of the host rock may melt when it breaks off from the because the product of a vector (here force) and a scalar (here the remain part of the roof or the walls, and subsides into the chamber ciprocal of area) is a vector (and, also, because the orientation of the (a process referred to as stoping). surface on which it acts is assigned a priori). At a point in the earth's Many hydrocarbon reservoirs are compartmentalised, that is, are crust, however, there act numerous stress vectors, and the three- composed of domains or parts (compartments) that are mechanically dimensional collection of all these vectors, for the (fracture or contact different from the adjacent parts or domains (Economides and Nolte, or imaginary) planes of all possible attitudes at that point, define the 2000; Satter et al., 2008). For hydrocarbon reservoirs, compartments stress tensor. Thus, the state of stress at a point in the earth's crust is a are commonly domains with different pore-fluid pressures, so that second-order tensor, or simply a tensor, the orientation of the surface the pressures in adjacent parts of the reservoir may be quite different. on which it operates not being assigned a priori. The component of The compartmentalisation is often related to tight faults, that is, faults stress in any particular direction, however, is a vector. In physics, en- with very low permeability that separate different parts of the reser- gineering, and particularly in geosciences it is common to refer to the voir so that there is no fluid flow between the adjacent parts. stress in a given direction, such as the stress on a fault plane or stress Totally fluid magma chambers composed of a single magma type on a crystal slip plane. In this paper, the term traction is not used. In- cannot normally be compartmentalised. This follows because com- stead, the word stress is used both for the stress vector and the stress partments in the present sense can only arise if there are some phys- tensor. Also, compressive stress is regarded as positive and tensile ical boundaries, such as contacts or faults, that hinder the free flow of stress as negative, as is common in geosciences. material (here fluids) in response to pressure/hydraulic gradients be- The geostatic stress, or vertical stress, is given by: tween different parts of the chamber. However, many magma chambers contain magmas of very different compositions with widely z σ ¼ ρ ðÞ ð Þ different thermal and mechanical properties. Furthermore, the v ∫ r z gdz 2 magmas in a chamber are generally at various stages of solidification 0 and thus at a different temperature and viscosity. For example, on the top of a totally molten basaltic magma chamber there may be partial- for the case of crustal layers with different densities, that is, ρ(z), or ly molten, or solidified, acid magma that may be injected by basaltic as: dykes and inclined sheets. Basaltic sheets, for example, dissect some σ ¼ ρ ð Þ of the granophyre layers in the fossil chamber of Slaufrudalur v rgz 3 (Fig. 5). It is not known exactly at what time these injections took place, but some of them may have been intruded while the grano- for the case where all the crustal layers have the same density, name- phyres magma was still partially molten and behaving as poroelastic. ly ρr. Here, the vertical coordinate z is positive downwards, that is, to- In the latter case, the basaltic sheets could have formed barriers for wards increasing crustal depth, and g is the acceleration due to lateral fluid transport between nearby granophyric parts, thereby for- gravity. Geostatic stress is also referred to as overburden pressure ming compartments. which, however, is more appropriately identified with lithostatic Similar compartments, but on a much smaller scale, are common in stress (or pressure). The three concepts, geostatic stress, overburden magma chambers and exemplified by net-veined complexes (Fig. 10). pressure, and lithostatic stress/pressure, are used interchangeably in Any rock bodies with a low permeability and different mechanical prop- the geoscience literature. Here I shall mainly use lithostatic stress erties from the adjacent parts of the magma chamber may function as a with the following definition. temporary or permanent barriers to fluid transport and mixture, and For any stress state at a point in the crust, there are three mutually thus contribute to the generation of compartments (a detailed discus- orthogonal planes that are free of shear stress. These are named the sion of magma-chamber compartments is in Section 7). principal stress planes and the normal stresses that act upon these planes are known as the principal stresses. The three principal stress-

4. Crustal stresses es are denoted by σ1, σ2, and σ3, with the maximum compressive principal stress being σ1, the intermediate principal stress being σ2, Stress in is a measure of the intensity of force per unit area. That is, and the minimum principal compressive (maximum tensile) princi- the greater the intensity of the force for a given area on which the pal stress being σ3 (e.g., Jaeger et al., 2007; Gudmundsson, 2011a). 28 A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41

Fig. 10. Large-scale contacts between plutons and their host rocks tend to be smooth, as indicated by the roof of a part of the Austurhorn Pluton, a fossil magma chamber in South- east Iceland (Blake, 1966; Furman et al., 1992). The same plutons may show evidence of very complex small-scale interaction between widely different magmas, forming small- scale ‘compartments’ of so-called net-veined complexes. (a) Part of the roof of the Austurhorn Pluton, here exposed in the mountain Krossanesfjall (its height is 716 m). Also indicated are the comparatively smooth contact between the pluton and its roof and the contact between the primarily felsic and primarily mafic intrusions in part of the mountain. On the sandy coast, a part of the net-veined complex in Krossanesfjall is seen. (b) Close-up of the part of the complex seen on the flat ground close to the road in the photograph in (a). The very light-coloured rocks are felsic, the grey to dark rocks being mafic.

If all the principal stresses are equal (isotropic or hydrostatic or dykes, inclined sheets, or sills. If the excess pressure decreases, that is, spherical state of stress) and the stress magnitude increases with if the chamber volume decreases, the chamber shrinks (resulting in depth as in Eq. (3), then the state of stress is referred to as lithostatic. adeflation), so as to establish close-to-lithostatic pressure again. While deviation from lithostatic stress is associated with unrest pe- Thus, unless some considerable inflow (replenishment) or outflow riods, lithostatic stress is usually used as a reference state of stress of magma or other pressured fluids is taking place, the long-term me- for magma chambers. This follows because, except during unrest pe- chanical condition of a magma chamber is that of lithostatic equilibri- riods, the magma-pressure is likely to be close to the pressure or um with the host rock. stress in the host rock, that is, to be in a lithostatic equilibrium. For a host rock that behaves as elastic, as the earth's upper crust generally 5. Stresses and pressures associated with dykes and sills does during unrest periods (e.g., Mogi, 1958; McTigue, 1987; Dzurisin, 2006; Segall, 2010), the host rock responds immediately to The lithostatic equilibrium of a magma chamber may become dis- pressure changes in the chamber. If the pressure increases, that is, if turbed during external and/or internal loading, as normally happens there is an excess pressure (pressure above lithostatic—see a more during unrest periods. Loading here means forces, stresses, pressures, detailed definition in Section 5) in the chamber, the chamber expands or displacements acting applied to a body such as a magma chamber. (resulting in an inflation). If, eventually, the excess pressure reaches As mentioned, most magma chambers originate from sills and many the tensile strength of the host rock the chamber ruptures and injects remain sill-like throughout much of their life-times (Figs. 7d, 8, 9). A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41 29

Also, sills are normally fed by dykes. It is therefore appropriate to start our analysis of stress around magma chambers by considering stresses around sills and dykes. Consider first the emplacement of a sill, that is, the deflection of a dyke into a sill (Figs. 3, 8; cf. Gudmundsson, 2011b). For a pure hydrofracture such as a dyke, the fracture path is opened by overpressure (driving pressure, net pressure) of the magma. It is this overpressure that drives all hydrofractures open, including man-made hydraulic fractures used for crustal stress measurements and in the geothermal and petroleum industries to increase the permeability of reservoirs (e.g., Hubbert and Willis, 1957; Sneddon and Lowengrub, 1969; Sun, 1969; Daneshy, 1978; Warpinski, 1985; Spence et al., 1987; Lister and Kerr, 1991; Rubin, 1995; Valko and Economides, 1995; Amadei and Stephansson, 1997; Yew, 1997; Economides and Nolte, 2000; Zoback, 2007; Zang and Stephansson, 2010). For the dyke to form and propagate, the magmatic overpressure must be sufficiently high to overcome the tensile strength of the host rock T0 and the normal stress on the magma-filled fracture, that is, the dyke fracture. Field observations, primarily cross-cutting relations, show that most dykes and sills (and inclined sheets) are extension fractures. This means that the fracture forms in a plane that contains two of the principal stresses, namely the maximum and the intermediate stresses σ1 and σ2. That plane, by definition, is perpendicular to the minimum principal stress σ3 (Fig. 11). When a dyke forms, the state of stress is normally anisotropic, that is, non- lithostatic, so that the principal stresses are unequal. Because most dykes are emplaced in rift zones, the normal state of stress is such that σ3 is sub-horizontal and perpendicular to the strike dimension of the forming dyke, σ2 is also sub-horizontal and parallel with the strike of the dyke, and σ1 is sub-vertical and parallel with the dip dimension of the dyke (Fig. 11). This state of stress is the normal one Fig. 11. State of stress encouraging the emplacement of a dyke. The vertical stress is the σ during rifting events at divergent plate boundaries. maximum principal compressive stress, 1, the horizontal stress parallel with the σ σ trend/strike of the dyke is the intermediate principal compressive stress, 2, and the The vertical stress 1 is given by Eqs. (2), (3) and clearly includes horizontal stress perpendicular to the dyke is minimum principal compressive (maxi- the effects of the acceleration due to gravity g. In fact, the vertical mum tensile) stress, σ3. Here, a 2.5-m thick basaltic dyke from the Quaternary lava pile stress given by Eq. (3) follows directly from the acceleration due to (a palaeorift zone) in Southwest Iceland. When the dyke becomes emplaced, it may, gravity, g, and the definition of stress as given in Eq. (1). In Newton's temporarily, change the normal stress field so as to encourage the subsequently fl second law of motion, force is defined as mass times acceleration emplaced dyke to become de ected into a sill (Figs. 1, 3, 8). (F=m×a). Close to the Earth's surface, the acceleration a in the − 2 Earth's gravity field is g (on average, 9.81 m s ), in which case the assumes that the horizontal strains (here ε3 =ε2) in the host rock are second law may be written as: zero. When that is not the case, Eq. (6) is commonly rewritten as (Fyfe et al., 1978):

F ¼ m g: ð4Þ σ σ ¼ 1 ε E ð7Þ 3 m−1 3

where E is Young's modulus. The mass m of a rock particle or body is equal to the body's volume Eqs. (6) and (7) make many assumptions, most of which are listed times its density. For a rock column of unit cross-sectional area, say by Gudmundsson (2011a, p. 112). The equations be applied to esti- close to a dyke (Fig. 12), the volume must be the unit area A times mate the tensile stress close to a totally molten magma chamber, or the crustal depth z. With A=1, the volume is z (in units of m3), and other fluid-filled reservoirs, because, using a common value of we have: Poisson's ratio of 0.25 (Carmichael, 1989; Hansen, 1998; Bell, 2000; Myrvang, 2001), the minimum theoretical principal stress would be F mg ρ zg about one-third of the maximum principal stress, or: σ ¼ ¼ ¼ r ¼ ρ gz: ð5Þ 1 A A 1 r 1 σ ¼ σ ð8Þ 3 3 1 Here the vertical stress (for a rift zone) is assumed to be the max- and such a stress difference is normally not possible to reach in the vi- imum principal stress σ . In an extensional regime, such as at a diver- 1 cinity of a rock-magma (or other fluid) contact. For a magma chamber gent plate boundary or a rift zone, the horizontal stress (here σ )is 3 initially in a lithostatic equilibrium, long before σ could be reduced related, theoretically, to the vertical stress (here σ ) through the fol- 3 1 to 1/3σ , a hydrofracture (a dyke, an inclined sheet, or a sill) would lowing equation (Fyfe et al., 1978): 1 be injected and the stress difference reduced (Gudmundsson, 2006). νσ σ Eqs. (6)–(8) together with Eqs. (2)–(5) show that the effect of σ ¼ 1 ¼ 1 ð6Þ 3 1−ν m−1 gravity, through the acceleration g, is always included in the calculations. As a consequence, most authors do not consider the total pres- where ν is Poisson's ratio, m=1/ν is Poisson's number (the reciprocal sure in a magmatic intrusions or a magma chamber when analysing of Poisson's ratio), and the other symbols are as defined above. Eq. (6) the stresses around these structures but rather the excess pressure 30 A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41

along the dyke path even though the excess pressure at the ﬂuid source is normally equal to the rock tensile strength and thus only several megapascals.

(3) Total pressure (pt) in a magma chamber is its excess pressure plus the lithostatic stress (or overburden pressure). When a magma chamber is in a lithostatic equilibrium with its host rock, that is, in the absence of unrest, there is normally no tectonic activity associated with the chamber, such as hydrofracture initiation or faulting. It is only when there is some (usually positive, but occasionally negative) excess pressure in the reservoir that stresses build up in the surrounding rocks which, eventually, may result in faulting or hydrofracture initiation. The conditions for hydrofracture initiation from a ﬂuid source are sometimes given using the total pressure, namely as (Jaeger et al., 2007):

¼ σ þ ð Þ pt 3 T0 9

where T0 is the in situ tensile strength of the host rock, and the other symbols are as deﬁned above. Eq. (9) may be used for dykes (and sills and inclined sheets), but is most useful for the initiation of hydraulic fractures injected from drill holes (Valko and Economides, 1995; Amadei and Stephansson, 1997; Yew, 1997; Economides and Nolte, 2000; Zoback, 2007; Zang and Stephansson, 2010). Because the total pressure is equal to the excess pressure plus the lithostatic pressure, we have:

¼ þ ð Þ pt pl pe 10 Fig. 12. State of stress in the crust close to a dyke, with reference to Eqs. (2), (3), (5), and (12). The unit cross-sectional area of a rock column used is indicated, as well as so that, using Eq. (9), we can write Eq. (10) in the form: the differential stress σd at the crustal level where the dyke is measured, and used in Eq. (12). The depth of the unit area below the earth's surface at the time of dyke em- þ ¼ σ þ ð Þ placement is z in Eqs. (2), (3), and (5), whereas the height of the dyke exposure pl pe 3 T0 11 above the magma source (the dyke dip dimension) is h in Eq. (12).

where pl is the lithostatic stress at the rupture site in the walls of the magma chamber, p is the total magmatic pressure in the chamber, σ or the overpressure. The main pressure concepts may be defined as t 3 is the minimum principal stress, and T the local in situ tensile follows (Gudmundsson, 2011a): 0 strength at the rupture site. If a dyke (or an inclined sheet) becomes injected into the roof of the chamber and starts to propagate up (1) Excess pressure (p ) in a magma chamber (or other e into the crustal layers above the chamber, the magmatic overpressure hydrofracture sources) is the magma pressure in excess of p in the dyke is given by: the lithostatic pressure (or overburden pressure), that is, the o total pressure (defined below) minus the lithostatic pressure. p ¼ p þ ðÞρ −ρ gh þ σ ð12Þ Excess pressure at the time of rupture and hydrofracture for- o e r m d mation is normally equal to the tensile strength of the host where ρ is the average host-rock density, ρ is the average magma rock of the reservoir and thus generally in the range of r m density, g is acceleration due to gravity, h is the dip dimension or 0.5–6 MPa, with a maximum of about 9 MPa, and most com- height of that part of the dyke above the point of rupture and dyke monly about 3 MPa. initiation, and σ is the differential stress (σ =σ −σ ) at the level (2) Overpressure (p ) – also named driving pressure and net pres- d d 1 3 o where the dyke is examined (Fig. 12). sure – is the pressure that drives the propagation of a Eq. (12) can be used to estimate the likely overpressure of a dyke hydrofracture (a fluid-driven extension fracture), such as a that has reached a certain elevation (height or dip dimension) in the dyke, a sill, or an inclined sheet. Overpressure is the result of crustal layers above its magma chamber (Figs. 1, 3, 11–14). When the combined effects of the initial excess pressure in the using Eq. (12), however, the following should be considered: magma chamber and the magma buoyancy, the latter being due to the difference between the density of the fluid in the (1) At the time of dyke initiation from a reasonably large magma

fracture and the density of the rock through which the fracture chamber, the excess pressure pe =pm −pl is, as a rule, positive, propagates. It is the total pressure minus the normal stress as is needed to rupture the chamber walls. Normally, when the which acted on the potential sheet (dyke or sill) fracture before excess pressure reaches roughly the tensile strength, in which

magma emplacement; for a hydrofracture the normal stress is case pe =T0, the reservoir ruptures in tension and a dyke (or an the minimum principal compressive stress σ3. The excess pres- inclined sheet or a sill) initiates. sure decreases along the flow direction in the fracture: for ex- (2) The differential stress, defined as σd =σ1 −σ3, is either zero or ample, up the dip-dimension of a sub-vertical dyke. By positive (Fig. 12); it cannot be negative because σ1 ≥σ2 ≥σ3 so contrast, the buoyancy term increases so long as the average that σ1 cannot be less than σ3. When σd =0, then σ1 =σ3 so density of the host-rock through which the fracture propagates that the state of stress is isotropic (here in two dimensions) is greater than the density of the fluid (here magma). The over- and when applied to three dimensions (so that we also have

pressure may reach several tens of megapascals at some point σ1 =σ2), then the state of stress is lithostatic. A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41 31

Fig. 13. High-density basaltic feeder-dyke passing through many low-density pyroclastic rocks (some layers are indicated) in the caldera wall of Las Canadas in Tenerife (Canary Islands). View east, the caldera wall is close to 300 m high. Part of the same feeder dyke is seen in Fig. 4b. Several other basaltic dykes and sills are seen in the section.

(3) The density difference ρr −ρm can be (1) negative, when the Since dykes injected from magma chambers feed most eruptions, magma is denser than rock; (2) zero, when the density of the let us brieﬂy consider the conditions for feeder-dyke formation and, magma is equal to that of the rock; or (3) positive, when the in particular, the effect of ‘neutral buoyancy’ on the chance of the rock is denser than the magma. The density of most crustal dyke reaching the surface. For dyke initiation, Eqs. (9) or (11) must −3 rocks is in the range of 2000≤ρr ≤3000 kg m (Carmichael, be satisﬁed. At its initiation, there is no buoyancy effect on the 1989; Hansen, 1998; Bell, 2000; Schon, 2004) whereas that of magma pressure because h, the height or the dip dimension of the −3 typical magmas is mostly in the range 2250≤ρr ≤2750 kg m dyke as given in Eq. (12), is zero. However, as the dyke propagates (Murase and McBirney, 1973; Kilburn, 2000; Spera, 2000). upwards into the crustal layers above the roof of the chamber, its dip dimension h gradually increases, so that buoyancy, as presented When dealing with stresses around magma chambers and dykes by the second term on the right-hand side of Eq. (12), adds to its ini- and other sheet intrusions, it is the excess pressure (for the chamber) tial excess pressure pe so as to generate the overpressure po. The and the overpressure (for the dyke) that is important for brittle defor- buoyancy effect depends on the density difference between the mation. The total pressure is rarely used. It follows from the analysis magma and the host rock. If they are of equal density, on average, and equations above that the effects of gravity are automatically then the buoyancy effect remains zero. If the magma is denser than taken into account in such an analysis. the average density of the crustal layers though which the dyke

Fig. 14. Many basaltic dykes pass easily through low-density pyroclastic rocks (some layers are indicated) in the peninsula of Anaga, Tenerife (Canary Islands). Comparatively low density of layers, or a density similar to that of the magma (‘neutral buoyancy’), is normally not sufficient to arrest propagating basalt dykes. Dykes 1 and 2 are 1–2 m thick, and dykes 3 and 4 both around 5 m thick. A black car close to Dyke 3 also provides a scale. 32 A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41 propagates, then the buoyancy effect is negative, so that the excess the surface. In fact, as indicated above, the top crustal layers at all di- pressure at the rupture gradually decreases with height above the vergent plate boundaries worldwide have densities less than those of chamber. If, however, the rock is denser than the magma, then buoy- typical basaltic magmas. Nevertheless, basaltic dykes or sheets pass ancy effect is positive and the magmatic overpressure increases with through these layers to reach the surface in every single basaltic erup- increasing dip dimension h as the dyke propagates towards the tion. Thus, clearly, dykes do not as a rule change into sills at levels of earth's surface. neutral buoyancy; and neutral buoyancy layers/units do not stop the The average density of the uppermost crustal layers through vertical propagation of the dykes. which a dyke (or an inclined sheet) propagates is commonly similar Mechanically, there are no particular reasons why dykes should to or less than that of a basaltic magma (Figs. 13, 14). And, generally, stop, change into sills, or propagate laterally at levels of neutral buoy- the average density of the uppermost several hundred metres of the ancy. From Eq. (12) it follows that, for a gradually increasing average crust of a volcano-tectonically active rift zone is everywhere less host-rock density with increasing crustal depth (as is commonly than that of typical basaltic magma. The basaltic magma may have crudely the case), the highest magmatic overpressure occurs at the densities between 2600 kg m− 3 and 2750 kg m− 3, whereas the up- ‘regional’ level of neutral buoyancy. So, unless the local stress field, permost crustal layers may have densities as low as 2500 kg m−3, the tensile strength, or toughness of the rock change abruptly at the even in a predominately basaltic crust (e.g., Gudmundsson, 1988, contact with the level of neutral buoyancy, there is all the reason 2011a). Thus, to reach the surface, basaltic magma almost always for the dyke to continue its subvertical propagation path—that is, to has to propagate through crustal layers of densities that are less propagate though the level of neutral buoyancy. And this is, indeed, than that of the magma. The basaltic magma has normally to propa- what is observed in the thousands of basaltic dykes that propagate gate through many layers where the buoyancy is zero, so-called though layers of much lower densities, such as layers of basaltic brec- ‘levels of neutral buoyancy’, or even negative (Figs. 13, 14). This fol- cias (hyaloclastites), ignimbrites, and rhyolites (Figs. 4, 13, 14). lows because typical rift zones are composed of a variety of rocks. Even a predominantly basaltic crust usually contains numerous layers 6. Stresses around magma chambers of breccias, pyroclastics (hyaloclastites), sediments, intrusions and other rocks with different densities (Figs. 13, 14). Thus, for a basaltic 6.1. Stresses at magma-chamber initiation dyke propagating through such a pile, there are normally many layers of densities equal to that of the magma at a particular location (the We have considered the deflection of a dyke into a sill and seen magma density also changes with elevation in the dyke/conduit be- that when we work with overpressure (net pressure, driving pres- cause of degassing and other processes). Generally, therefore, there sure) the analysis automatically considers the effects of gravity, as in- are many ‘levels of neutral buoyancy’ for a basaltic magma on its dicated in Eqs. (2)–(7). For a sill (or a dyke) to form, the conditions of way to the surface (Fig. 13). Eqs. (9) or (11) must be satisfied. For a sill, the minimum principal

A ‘level of neutral buoyancy’ has often been suggested as a trap for compressive stress, σ3, is vertical and given by Eqs. (2) or (3). The basaltic magma, so as to either arrest dykes or deﬂect them into sills— stress is the same on the roof of the sill as on the rocks just ahead of and thereby generate potential magma chambers (Bradley, 1965; its tips. Similarly, for a magma chamber in lithostatic equilibrium, Holmes, 1965; Gretener, 1969; Francis, 1982; Ryan, 1993; Chevallier the vertical stress on the roof of the chamber is the same as that, at and Woodford, 1999). As we have seen, however, basaltic dykes the same crustal level and for the same rock density, just outside pass easily through layers of densities less than those of typical basal- the roof. tic magmas (Figs. 4, 13, 14). This is seen everywhere in the world Consider the fossil magma chamber in Fig. (5a). If the vertical where basaltic volcanism takes place. Most of this volcanism is sup- stress on the roof of the chamber at, say, point A is given by Eqs. (2) plied with magma through dykes, and all these dykes must pass and (3), then, if point B is at the same crustal depth as point A and through (usually many) ‘neutral buoyancy’ layers on their paths to the rock density in the columns above these points is the same (and the chamber is in lithostatic equilibrium), the vertical stress at point B must be the same as at point A. If the magma density equals the rock density (which may or may not have been the case in Fig. 5), then it follows that the increase in vertical stress with depth from point A is also given by Eq. (3) and the same applies to the vertical stress along the chamber wall below point B. Lets for a moment see what the consequences would be for a sill formation or magma-chamber development if the vertical wall- parallel stress at B and parallel with the wall of the fossil chamber was much higher, say double, the vertical stress on the top of the magma chamber. Now the exposed fossil chamber is composed of sills (Fig. 5b), so that the top part in Fig. (5a) is a sill. In this scenario,

we have 2σv, that is, for the sill, 2σ3, acting on the rock at point B but σ3 at point A (Fig. 5b). The sill had to propagate from its feeder dyke (Figs. 3, 8), so that point B is always in the rock-wall just ahead of the

propagating sill tip. This means that the vertical stress σv must suddenly change from σ3 on the top (the roof) of the sill to 2σ3 just ahead of the tip of the sill. But this would mean that the sill could not propagate at all. On the Fig. 15. Compartments in a magma chamber may be generated through faulting, for example the formation of (here nested) collapse calderas (for the formation of nested cal- top of the sill the vertical stress, given by Eq. (3), is in balance with deras see Geyer and Marti, 2009). Flow of magma between compartments 1 and 5, and the magmatic pressure in the sill and the opening or aperture of the 2 and 4 is unlikely since the magma would then first have to flow through compart- sill is entirely due to its overpressure as given, for example, by ment 3. Density differences between the magma in the upper part of the chamber Eq. (12). These results are well know elastic crack theory (Sneddon and in its lower part make such a flow unlikely. For the same reason, flow between and Lowengrub, 1969; Gray, 1992; Tada et al., 2000) and confirmed compartments 1 and 2, and 5 and 4 is likely to be minimal if any. The magma in each of the compartments may therefore, for a while at least, evolve largely independently by numerous hydraulic fracture experiments worldwide (Valko and of the magmas in the other compartments. Economides, 1995; Amadei and Stephansson, 1997; Yew, 1997; A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41 33

Economides and Nolte, 2000; Zoback, 2007; Zang and Stephansson, rock stresses normal to its surface (walls) at every point along the 2010). So it is only the overpressure that drives the sill. This overpres- surface of the chamber. Since it is further assumed that the chamber sure has only to overcome the tensile strength of the rock, according is fluid (a free surface), it follows that at the surface of the chamber to Eqs. (9)–(11). the principal stresses are all equal in magnitude and either parallel For a typical basaltic sill, the overpressure at a few kilometre depth or perpendicular to the chamber. These stresses (more specifically, might be 10–20 MPa. Take, as an example, a sill emplaced at the the stress vectors) must also, by definition, all be equal to the total depth of 3 km in the crust, a depth where many shallow magma fluid pressure (the lithostatic pressure) in the chamber at every chambers form. For the upper part of a typical crust at a divergent point on its surface. plate-boundary, the rock density is, on average, about 2600 kg m− 3 On the assumption of lithostatic state of stress, for a chamber in

(Gudmundsson, 1988), in which case the vertical stress, from lithostatic equilibrium (‘lithostatic conditions’ [with] ‘ρr =ρm’), as is Eq. (3) is about 76 MPa. For a sill, the overpressure is the total pres- assumed in Fig. 5 of the EWS-model, there cannot be any difference sure minus the minimum principal compressive stress σ3, here the between the ‘across wall’ (radial or normal), ‘wall-parallel’ (tangen- vertical stress σv according to Eq. (3), namely 76 MPa. If the tial or circumferential) stresses, and the vertical stress. Nor can overpressure is 20 MPa, then the total magmatic pressure in the sill there be any difference between any of these stresses at any point is 96 MPa. at the surface of the chamber and the total (lithostatic) pressure in If total vertical stress at the lateral tip or edge of the sill suddenly the chamber. From these assumptions it also follows that the increase changes to 2σ3, then, in this case, it becomes 152 MPa. Since in total pressure in the chamber with depth is exactly balanced by the 96–152=−56 MPa, it follows that the potential overpressure is less increase in the host-rock stresses on the chamber surface (or walls). than zero, that is, negative (−52 MPa), so that there is absolutely Thus, from Eqs. (2) and (3) and the assumptions above it follows no overpressure in the sill, which means that it could not form in that the lithostatic stress in the host rock and the pressure in the the first place, let alone propagate to form a magma chamber. This chamber at any depth z is the same, namely ρrgz=ρmgz. This means brings us to a recent model on the stress field around magma that the stress below points A and B in Fig. (5a) is the same at any chambers that includes extra wall-parallel stress. depth under consideration. Under these conditions, the effective stress (the stress giving rise to brittle deformation such as magma- 6.2. Different approach to magma-chamber stress modelling chamber rupture and dyke injection) is zero, so that there is no ten- dency to large-scale brittle deformation of any kind. For static fluid The authors of several recent papers (Grosfils, 2007; Hurwitz et al., pressure in a magma chamber, the mean stress is the hydrostatic 2009; Long and Grosfils, 2009) have suggested that standard analyti- pressure and we might thus also refer to the deviatoric stress, that cal elastic solutions for stress fields around magma chambers, such as is, the difference between the normal stress on a plane and the those by Davis (1986), McTigue (1987), Tait and Jaupart (1989), De mean stress on that same plane, when discussing the conditions for Natale and Pingue (1993), Saunders (2001), Gudmundsson (2002, brittle deformation around a magma chamber (Gudmundsson, 2006), Pinel and Jaupart (2003), and Masterlark (2007) as well as 2011a). In volcanotectonics, the effective stress may be regarded as similar viscoelastic solutions (Bonafede et al., 1986; Folch et al., the excess pressure, pe (Eqs. (10), (11)); which is the pressure that 2000; Trasatti et al., 2005) are ‘correct but incomplete’. The solutions can give rise to magma-chamber rupture. proposed by the authors themselves differ from those of almost all There is less-than-lithostatic fluid pressure in many porous hydro- previous solutions in three important respects, namely as regards carbon reservoirs. In fact, many have fluid pressures less than hydro- the following points (e.g., Grosfils, 2007): static although some reach lithostatic pressures (Chilingar et al., 2002; Satter et al., 2008). But these are not totally fluid reservoirs, (1) They add an extra ‘wall-parallel component of the lithostatic as is assumed in the EWS-model (and many other magma-chamber stress’ which is also supposed to ‘resist rupture’ of the stress models), but rather porous and fractured rock bodies that are magma chamber walls (and thus dyke, sheet, or sill injection partially filled with fluids (oil, gas, and water). A totally fluid reservoir from the chamber). It follows that there is ‘an important of any kind, such as the magma chambers discussed here, always depth-dependent factor which also resists the rupture process’. seeks to be in a lithostatic equilibrium with the host rock at any (2) In some of their models the tensile strength of the rock hosting point. If the reservoir is composed of pressure compartments (Fig. the magma chamber is taken as zero at the point of failure. This 15; see Section 7), so that the magma excess pressure may be higher implies that the magma excess pressure at that point must be in one part of the chamber than in other parts, the chamber would re- zero. spond through expansion in high-pressure part and eventually, if the (3) They claim that ‘a magma chamber can support a wide range of excess pressure reached the conditions of Eqs. (9)–(11), rupture and uniform pressures P prior to the initiation of tensile failure. inject dykes or sheets. Thus, even if there are differences in the densi- This result is certainly at odds with studies that claim that fail- ties of the host rock and the magma, so that, ρ ≠ρ , and gradients in ure will occur when P rises to at most only a few MPa above the r m stress and excess pressure, the chamber would tend to reach an equi- lithostatic’. In Table 1 of Grosfils (2007) the P-values in magma librium geometry with the host rock. Effects of stress gradients on the chambers before failure [uniform pressures; here excess mag- geometry of fluid-filled crustal fractures are discussed by Secor and matic pressures before rupture] are predicted to range up to Pollard (1975) and Pollard (1976). hundreds of megapascals. It follows from these considerations that there cannot be any extra Here we discuss these three main points and the associated model, ‘wall-parallel component of the lithostatic stress’ that resists magma- referred to as the EWS (extra wall-stress) model. All the references chamber rupture. If such a component existed, it would imply that are to the principal paper by Grosfils (2007) unless otherwise stated. the conditions σ1 =σ2 =σ3 (or σr =σθ =σz) were not satisfied, in direct contradiction with the definition of lithostatic state of stress. For 6.2.1. The first point the assumed lithostatic state of stress, the conclusions of the In the EWS model, the first reference state of stress is lithostatic, EWS-model as to the first point above are thus incorrect. that is, σ1 =σ2 =σ3 or, in the EWS-model notation σr =σθ =σz, and The second state of stress considered in the EWS-model is the the stresses increase in magnitude according to Eqs. (2) and (3). For well-known state of stress presented by Eqs. (6) and (8). It is often re- this reference state, it is assumed that, in the absence of unrest, the ferred to as the condition of uniaxial strain. The application of this magma chamber is in lithostatic equilibrium with its host rock. This stress state to geological situations is briefly discussed in connection implies that the pressure in the magma chamber equals the host- with Eq. (8) and in detail by Gudmundsson (2011a). The main 34 A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41 limitation, in the present context, is that this state of stress can never 7. Chamber pressure variation during an eruption: effects exist in the vicinity of a totally fluid magma chamber; normally, as of compartments soon as the stress difference σz −σθ (in the EWS-model notation) reached the tensile strength of the rock at any point at the surface The stress condition for magma-chamber rupture can be reached of the magma chamber, the chamber would rupture and a dyke or a in two basic ways (Gudmundsson, 1988, 2006; Folch and Marti, sheet or a sill would be injected. The fluid overpressure of the 1998): (i) through increasing the total pressure inside the chamber magma-driven fracture (Eq. (12)) would lessen the stress difference (for example, by adding magma to the chamber or through gas exso- so as to bring the state of stress close to lithostatic. This second lution from its magma), and (ii) through external extension, such as state of stress is thus not appropriate when considering totally fluid in rift zones, where the divergent plate movements gradually reduce magma chambers. That it is inappropriate is supported by the EWS- the minimum principal compressive stress σ3. Both loadings increase model results themselves which indicate that ‘even when the the chamber excess pressure. The second type of loading (ii) general- magma pressure P=0 [i.e., the excess pressure in the chamber is ly favours the injection of vertical dykes, whereas the first type of zero] the spherical cavity [the magma chamber] will fail when loaded loading (i) may sometimes favour dykes, and sometimes inclined by the weight of the enclosed magma alone, no other pressure com- sheets (and occasionally sills). Both loadings result in the increase ponent is required.’ This would mean, of course, that no such of the magma-chamber excess pressure. magma chamber could form in the first place. Once the excess pressure in a magma chamber reaches the conditions of rupture (Eqs. (9)–(11)), a magma-driven fracture (a dyke or an inclined sheet or, more rarely, a sill) is initiated. The stress condi- 6.2.2. The second point tions for the propagation of the fracture to the surface, resulting in an When a totally fluid magma chamber is subject to a ‘pressure P eruption or, alternatively, the arrest of the fracture at depth in the which acts uniformly upon all the parts of reservoir wall’, then if volcano (Fig. 1), have been discussed in many papers (e.g., Spence any part of the wall or surface has zero tensile strength there cannot et al., 1987; Lister and Kerr, 1991; Clemens and Mawer, 1992; be any significant excess pressure in the chamber at that point. Such a Petford et al., 1993; Rubin, 1995; Gudmundsson, 2002; Rivalta et al., magma chamber could never give rise to any normal eruption since 2005; Acocella and Neri, 2009; Geshi et al., 2010, 2012; Maccaferri for an eruption to occur and be maintained over some significant et al., 2010, 2011; Moran et al., 2011; Taisne et al., 2011). I shall there- time – eruption durations typically range from a few days to a few fore not discuss the effect of magma-chamber stress fields on dyke months (Simkin and Siebert, 2000) – there must be excess pressure propagation but rather focus on the excess-pressure changes in the in the chamber to drive out the magma (see Section 7 for the chamber during the eruption. excess-pressure analysis). Also, the use of zero tensile strength nei- Once the feeder-dyke has reached the surface, the volumetric flow ther fits with the other EWS-model estimates of magma-chamber rate of magma (assuming laminar flow) through the volcanic fissure excess pressures (and thus tensile strengths) of hundreds of mega- Q is given by (e.g., Gudmundsson, 2011a): pascals, nor with measurements of in-situ tensile strengths. This brings us to the third point. 3 Δu W ∂p Q ¼ ðÞρ −ρ g sinα− e ð13Þ μ r m ∂ 12 m L 6.2.3. The third point The third point suggests that magma chambers can tolerate mag- where Δu is the opening or aperture of the feeder-dyke or volcanic matic excess pressures of hundreds of megapascals. In fact, the excess fissure, W is the length or strike dimension of the feeder-dyke (the fi μ pressure needed for failure at great depths is apparently supposed to volcanic ssure) at the surface, m is the dynamic (absolute) viscosity ρ ρ become close to 3-times the lithostatic stress. Laboratory and in-situ and m the density of the magma (assumed constant), r is the aver- tensile-strength measurements have been carried out over many de- age density of the crustal segment (including the volcano; Fig. 1) cades. The highest tensile strengths reported from laboratory mea- through which the dyke propagated to the surface, g is the accelera- α ∂ ∂ surements on small specimens are about 30 MPa (Carmichael, 1989; tion due to gravity, is the dip of the feeder-dyke, and pe/ L is the Hansen, 1998; Myrvang, 2001; Gudmundsson, 2011a and references vertical excess-pressure gradient in the direction of the magma therein). More appropriate for magma-chamber failure and dyke or flow, that is, in the direction of the dip dimension of the dyke L. sheet injection, however, are in-situ tensile strength measurements Eq. (13) follows directly from the Navier–Stokes equation (Lamb, using hydraulic fracturing. These are made in a section of a borehole 1932; Milne-Thompson, 1996) and has been used in various forms which is sealed off (by rubber packers). The fluid (usually water) for analysing magma transport to the surface (Wilson and Head, pressure in this part is increased until the walls of the borehole fail, 1981; Lister and Kerr, 1991; Rubin, 1995; Gudmundsson and forming a fluid-driven extension fracture (a hydraulic fracture). The Brenner, 2005). experiment is then repeated and the fluid-pressure difference be- The total volume V that flows out of the poroelastic magma cham- tween that at initial rupture (fracture formation) and new fluid injec- ber through the feeder-dyke before the eruption comes to an end can tion into the new hydraulic fracture gives the in-situ tensile strength be estimated as follows (Gudmundsson, 1987): (Amadei and Stephansson, 1997; Myrvang, 2001; Zoback, 2007; Zang ¼ β þ β ð Þ and Stephansson, 2010). The analogy with magma-chamber rupture V fpe p m V c 14 and dyke injection is clear.

Worldwide results show that the in-situ tensile strengths of crustal where f is porosity (magma fraction) of the chamber, pe is the magma rocks vary between 0.5 and 9 MPa, and are mostly 1–6MPa(Haimson excess pressure in the chamber before rupture and feeder-dyke for- and Rummel, 1982; Schultz, 1995; Amadei and Stephansson, 1997; mation, βm is the magma compressibility and βp the pore compress- Zang and Stephansson, 2010). The strengths have been measured ibility of the magma chamber, and Vc is the total volume of the from the surface to depths of about 9 km. There is no known general chamber. Eq. (14) follows from poroelastic considerations (cf. Bear, increase in tensile strength of crustal rocks with depth. Neither are 1972; Wang, 2000) and is widely applied in hydrogeology there any rocks known to tolerate tensile stresses of hundreds of (Domenico and Schwartz, 1998; Deming, 2002). A non-porous (total- megapascals before failure. The EWS-model conclusion that excess ly fluid) version of this equation has also been used for magma cham- pressures at magma-chamber failure may be as high as hundreds of bers (e.g., Machado, 1974; Blake, 1981). The flow out of the magma megapascals is thus without any support from direct measurements chamber through the feeder-dyke stops, and the eruption comes to of tensile strengths of crustal rocks. an end, when the excess pressure is no longer able to keep the A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41 35 dyke-fracture open at its contact with the chamber, that is, when form as the well-known equation for the relaxation of viscoelastic pe →0. and poroelastic bodies (e.g., Williams, 1980; Christensen, 1982; For dykes that do not reach the surface, that is, non-feeders, the Wang, 2000; Yu, 2000). For example, for Eq. (19) the only differences fracture need not close when the dyke stops its propagation. When are that for a viscoelastic Maxwell body, Young's modulus is the dyke tip becomes arrested, the dyke propagation simply stops substituted for the volumetric flow rate Q and the dynamic viscosity and the emplaced dyke solidifies and eventually cools down to the of the viscous part of the Maxwell body is substituted for the erupted temperature of the host rock. In rare cases where the roof of an old (and intruded) volume V (Williams, 1980). This analogy is as magma chamber is exposed (Fig. 5), dykes may be seen connected expected because the excess-pressure decrease in the chamber is for- with the magma chamber, particularly when the dyke magma is felsic mally analogous to the relaxation or decrease of stress in a viscoelas- and highly viscous. Most such dykes, particularly the comparatively tic body. thick ones, are presumably non-feeders. This follows because for the That the volumetric flow rates falls roughly exponentially during feeder-dykes, particularly basaltic ones, the contact with the magma an eruption is commonly observed (Machado, 1974; Wadge, 1981; chamber would normally effectively close at the end of the eruption. Thordarson and Self, 1993; Thordarson and Larsen, 2007). Similarly, Eqs. (13) and (14) can be combined to show theoretically how the roughly exponential decrease in excess pressure during an eruption excess pressure in the magma chamber during an eruption is likely to is probably a common feature in magma chambers. However, these decrease. Here we consider only basaltic eruptions. The simplest way conclusions rest not only on the assumptions given above, but also of showing this is to assume (1) that the volumetric flow rate over a on the assumptions that (1) the contribution of the exsolved volatiles given time t during the eruption can be expressed by some average and (2) replenishment (inflow of new magma into the chamber) value Qt and (2) that there is no density difference between the have negligible effects on the excess pressure evolution in the cham- host rock and the magma, so that the term (ρr −ρm)gh in Eq. (13) is ber during the eruption. The contribution of exsolved volatiles to the zero. This means that, for the present analysis, the magma transport excess pressure in the chamber has been discussed in detail by many up through the feeder dyke is entirely attributable to the excess- authors (e.g., Tait and Jaupart, 1989; Woods and Huppert, 2003), but pressure gradient, ∂pe/∂L. These assumptions are only made so as to is likely to be minimal for typical basaltic eruptions. The effects of re- make the mathematical analysis more tractable; the results still cap- plenishment on the excess-pressure evolution in the chamber have ture the essential physics describing the excess-pressure changes in also been considered by others (e.g., Woods and Huppert, 2003). a magma chamber during an eruption. However, for most eruptions, the volumetric flow rate out of the During flow of magma out of the chamber through a dyke, the ex- chamber during an eruption is many times larger than the rate of cess pressure pc in the chamber at any instant is given by: flow into the chamber (from a deeper source reservoir), so that this effect on the excess pressure variation is commonly minimal ¼ −ψ∫t ð Þ pc pe 0Qdt 15 (Stasiuk et al., 1993; Gudmundsson, 2006). From Eqs. (13), (14), (17), and (18), the volumetric flow rate or where pe is the excess pressure at the time of magma-chamber rup- effusion rate depends on the variation in the excess pressure in the ture, that is, at t=0, Q is the volumetric flow rate, and ψ is the recip- chamber, but also on other factors. From Eq. (13), these factors in- rocal of the right-hand side of Eq. (14), namely: clude the length of the feeder-dyke/volcanic fissure, the dynamic vis- hi cosity of the magma, the density difference between the magma and −1 p ψ ¼ f β þ β V ¼ e ð16Þ the host rock, the dip of the feeder-dyke, and the opening/aperture of p m c V the feeder-dyke/volcanic fissure at the surface. For a basaltic magma − fi and has the units of Pa m 3. of a given composition and viscosity, even if the length of the ssure The volumetric flow rate Q as a function of time may be given as may (and often does) change during the eruption, the main effect on fl (Machado, 1974): the volumetric ow rate would still be the variation in aperture. This is because the volumetric flow rate for laminar flow depends on the ¼ − ∫t ð Þ aperture in the third power – referred to as the cubic law – so that Q Q e A 0Qdt 17 small changes in the size of the aperture may have large effects on the volumetric flow rate (e.g., Wilson and Head, 1981; Stasiuk et al., where Qe is the initial volumetric flow rate and A is a constant that is related to Eqs. (13) and (14), in particular to the excess pressure and 1993; Gudmundsson and Brenner, 2005). the compressibility and volume of the reservoir, as well as on the di- Although many effusion rates show a gradual decline from a peak mensions of the feeder dyke. early in the eruption until it comes to an end (Machado, 1974; It can be shown that the solution to Eq. (17) is (Machado, 1974): Wadge, 1981; Stasiuk et al., 1993; Thordarson and Self, 1993; Thordarson and Larsen, 2007), there are commonly irregularities in ¼ −At ð Þ the volumetric flow rates. For example, the volumetric flow may in- Q Q ee 18 crease late in the eruption (Gudmundsson and Brenner, 2005), and For an eruption, the volumetric flow rate is also referred to as the the composition of the magma may change. The latter is very com- effusion rate. A similar equation was obtained by Wadge (1981).By mon in eruptions in stratovolcanoes and calderas where the initial analogy with Eq. (18), and using Eqs. (15) and (16), the excess pres- magma is the most evolved and becomes less so (more primitive or sure in the magma chamber after time t as a function of magma flow mafic) during the course of the eruption. out of the chamber (feeder-dyke plus eruptive materials) during that Variation in flow rates may, as indicated above, be partly related to time interval is (cf. Woods and Huppert, 2003): changes in feeder-dyke/fissure apertures as well as to density (hence buoyancy) changes of the magma during the course of the eruption. ¼ −ðÞQt=V ð Þ Changes both in flow rates as well as in composition, however, are pc pee 19 likely to be commonly related to processes inside the associated where all the symbols are as defined above and the exponent has the magma chamber. In particular, some changes in volumetric flow units of volume divided by volume and is therefore dimensionless. rate and composition may be related to magma-chamber compart- Eqs. (18) and (19) show that, for the given boundary conditions, ments and their interactions. the volumetric flow rate through the feeder dyke (eruptive fissure) As indicated above, compartments are well known in petroleum and the magmatic excess pressure in the chamber should decrease reservoirs (Economides and Nolte, 2000; Deming, 2002; Satter et al., exponentially during the eruption. These equations have the same 2008) but have received hardly any attention in volcanology. It is 36 A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41 well known, however, that nearby volcanic fissures in the same volca- Southeast Iceland, a fossil magma chamber, is thought to be composed no and, given their location, from the same magma chamber, separat- of at least 70 smaller intrusions (Fig. 16; Roobol, 1974). With an exposed ed by geologically short periods of time (tens or hundreds of years), area of around 20 km2, Vesturhorn, of late Tertiary age, is the largest plu- may produce eruptions of widely different duration and erupt mate- ton in Iceland and is located in a 900-m-thick pile of basaltic lava flows rials with a widely different composition. This indicates that there with a regional dip of 7° NW. At the contacts with the pluton, however, are nearby parts or volumes inside a magma chamber that have min- the dips of the lava flowsincreasetoasmuchas30°,mostlyawayfrom imal interaction (exchange of materials), that is, form different com- the pluton, indicating, again, that many magma chambers/plutons gen- partments within a single magma chamber. These are commonly erate space for themselves partly through forceful intrusion, primarily referred to as pressure compartments for hydrocarbon reservoirs by bending the layers above (and below). Part of the roof is preserved (Deming, 2002). They are generally explained in terms of structural (Fig. 16); it consists of flat-lying basaltic lava flows. Based on field map- constraints or boundaries, including faults and pressure seals, that ping, Roobol (1974) proposed that the exposed part of the pluton is com- to a large extent isolate one part of the reservoir from its other parts. posed of at least 70 different intrusions. However, the main parts of the An example of such a division of a magma chamber into compart- pluton are a ring complex, primarily of various types of gabbros and ments is through faulting, particularly following nested caldera (or granophyres, an eastern mafic complex, and several epigranite bodies. graben) formation (Fig. 15). Here the magmas at different elevations Similarly, the Austurhorn Pluton in Southeast Iceland is a large in the chamber are unlikely to mix laterally because the faults are bar- fossil magma chamber composed of many intrusions (Fig. 17; riers to the lateral fluid flow. For a density stratified magma chamber, Furman et al., 1992; Thorarinsson and Tegner, 2009). For example, the same magmas are unlikely to mix vertically. It follows that in the its main gabbro body (Fig. 17) is composed of at least 8 major units schematic scenario illustrated in Fig. 15 the chamber may be divided (Thorarinsson and Tegner, 2009). These units show clear evidence into compartments, such as those indicated by the numbers 1–5. Be- of having (i) received new magma injections (replenishment), fore and shortly after fault formation, the magmas in compartments 1 followed by periods of normal crystal and liquid fractionation, and 5 may have had very similar compositions, but subsequently they (ii) compositional variation with stratigraphic elevation within the could evolve differently (through fractionation and anatexis and exposed part of the chamber, and (iii) acted as a magma chamber stoping) so as to become quite different magmas. The same applies for a major stratovolcano. Within the Hvalnessfjall gabbro, the main to the magmas in compartments 2 and 4. Even if the faults were melt lens at any particular time may have been quite thin; ring faults, lateral flow is unlikely along large parts of the faults (so Thorarinsson and Tegner (2009) estimate it at 200–300 m. as to allow mixing between compartments, say, 90° apart or 180° Most magma chambers are not totally molten (except perhaps apart). This follows because lateral flow along fractures tends to be small sill-like chambers soon after their initiation), but rather a mix- perpendicular to the minimum principal compressive stress, σ3, and ture of melt and a crystal mush (e.g., Marsh, 1989; Sinton and Detrick, thus occupy that part of the ring fault that does not deviate much 1992; Marsh, 2000; Dobran, 2001; Parfitt and Wilson, 2008). Various from being roughly perpendicular to σ3. estimates have been made of the actual amount of melt in a body While faults and other ‘seals’ may contribute to the compart- identified as a magma chamber, but this amount is surely going to mentalisation of magma chambers, perhaps a more common reason vary a lot during the lifetime of the chamber. One geophysical esti- for compartments is the way magma chambers form. It is now mate for a basaltic magma chamber beneath the Juan de Fuca Ridge recognised that most magma chambers form through many magma in- (just offshore the west coast of North America) indicates that the ac- jections (Figs. 5, 8). For the fossil chamber of Slaufrudalur (Fig. 5)the tual melt in the chamber, located at a depth of about 2–6 km below contacts between the sills are still seen, so that the sills may have the ocean floor and with an estimated volume of 250 km3, is only acted as compartments with a minimal exchange of materials between 2–8% of the total chamber volume (West et al., 2001). them. Compartments in magma chambers exchange energy (heat) but There are, however, other ways for magma chambers to form gradu- commonly little if any matter (over long periods of time) and are ally over long periods of time. For example, the Vesturhorn Pluton in thus effectively closed thermodynamic systems. This implies that

Fig. 16. Part of the fossil magma chamber Vesturhorn in Southeast Iceland (its maximum height is 889 m). View northwest, the pluton is composed partly of felsic (granophyre) and partly of maﬁc (gabbro) bodies (some are indicated). Also indicated are a dyke and a sill dissecting the roof. The pluton has an exposed area of some 20 km2, is composed of more than 70 individual intrusive bodies of various sizes (Roobol, 1974), and is the largest pluton in Iceland. A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41 37

Fig. 17. Part of the fossil magma chamber Austurhorn in Southeast Iceland (its height as seen here is 606 m). View northwest, the pluton is composed of many smaller intrusions, of mafic and felsic composition (cf. Furman et al., 1992; Thorarinsson and Tegner, 2009). during eruptions the composition of eruptive materials issued from form. Most importantly, the local stresses around the chamber must nearby fissures fed by the same magma chamber may differ consider- be known if we are to forecast successfully magma-chamber rupture ably (Fig. 18). This applies, in particular, to small eruptions each of and dyke/sheet injection. When the conditions for chamber rupture which tends to draw its magma entirely from a single compartment are met, reliable stress models are needed to assess the likelihood (numbered 1–5inFig. 18). For larger eruptions, the magma is gradu- that the resulting dyke/sheet will reach the surface and supply ally drawn from deeper parts of the compartments (numbered 1a–3a magma to an eruption. Once an eruption has started, its duration de- in Fig. 18), whereby drawdown may lower the hydraulic potential so pends largely on the associated excess-pressure changes in the as to allow magma to migrate into the compartment from nearby chamber. compartments; for example from 2a and/or 3a to 1a during eruptions This paper provides a review and analysis of many current ideas from compartment number 1 (Fig. 18). and models, and presents some new ideas, as to the formation and There are thus likely to be some threshold values as regards low- geometric evolution of magma chambers. Field data and theoretical ering of the hydraulic potential (or pressure) in a compartment for considerations indicate that most magma chambers are formed magma from nearby compartments to start to migrate towards the through many magma injections; a few, presumably mostly small erupting compartment. The melt migration through the crystal sill-like chambers, are formed in single magma injections. These mush follows Darcy's law of fluid flow in porous media (Bear, 1972; McKenzie, 1984; Gudmundsson, 1987; Dobran, 2001; Bachmann and Bergantz, 2004). The flow is driven by the hydraulic gradient or, in the case of purely horizontal flow, by the pressure gradient. The flow rate depends on the permeability and porosity of the crystal mush, as well as on the viscosity and density of the melt and the cross-sectional areas of the channels through which it flows. When magma from an adjacent compartment is able to flow into the erupting compartment, the excess pressure necessary to continue the eruption may be maintained for a longer time (Eqs. 15–19). It follows that the duration of the eruption may be longer than if only the original compartment provided all the magma. Furthermore, the composition of the erupted magma is likely to change during the course of the eruption, as the new melt comes in from the adjacent compartment(s). If new melt is injected from a deeper source during the eruption, it is most likely to be of high density and may pond on the floor of the chamber (Fig. 18).

8. Discussion Fig. 18. Schematic illustration of possible compartments in a magma chamber. Com- – A magma chamber is normally a necessary condition for the for- partments 1 5 are composed of comparatively low-density magmas, whereas those number 7–11 are of high-density magmas (and, in this illustration, related to injection mation of a major, polygenetic volcano. To improve our understand- of primitive (basaltic) magmas from a deeper magma source). During typical small ing of the long-term behaviour and activities of stratovolcanoes, eruptions from compartments 1–5, magma is derived only from the compartment collapse calderas, and basaltic edifices, we must know the mechani- with which the feeder dyke is connected. For larger eruptions, magma is drawn from cal, and especially the stress, conditions for the initiation and evolu- deeper parts of the specific compartments (indicated by the numbers 1a, 2a, and 3a), and the associated lowering of the hydraulic potential may eventually result in tion of the associated magma chambers. In particular, we need to magma being driven into that compartment from one or more adjacent ones (for ex- understand the condition for the growth of a magma chamber from ample, driven from compartment 2 to compartment 1 during a comparatively large its nucleus, that is, the initial intrusion, to its quasi-stable geometric eruption from compartment 1). 38 A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41 views are elaborated in many recent papers and special issues, such Sparks et al., 1984; Jaupart and Tait, 1995; Spera and Bohrson, 2001; as by Menand et al. (2011). Examples of magma chambers generated Annen and Sparks, 2002; Walker et al., 2007), and these replenish- in many injections are given in Figs. 5, 8, 10, 16, and 17. Field data and ments affect the internal structure of the chamber and how it theoretical considerations also indicate that many magma chambers develops and responds to local stress fields. initiate from multiple sill injections of various shapes (Figs. 5, 8), whereas other multiple sills or sill clusters failed to evolve into cham- 9. Conclusions bers (Figs. 3, 4). These observations relate to the modelling of the state of stress around magma chambers. Some of the main conclusions of the paper may be summarised as While most analytical and numerical models on magma-chamber follows: stress field use the same basic assumptions as to how to model a fluid-filled chamber (e.g., McTigue, 1987; Tait and Jaupart, 1989; (1) Most magma chambers form and grow through repeated injec- Gudmundsson, 2002), a widely different approach has recently been tions of magma. Many chambers develop from sills, and some used by Grosfils and co-workers (Grosfils, 2007; the EWS-model) who retain their sill-like geometries throughout their lifetimes. suggest that the standard approach results in several incorrect results. (2) While active, a magma chamber acts as a sink for magma (re- Among the supposed-to-be incorrect results are (a) the presumed low ceives magma) from a deeper source (or sources), here re- magmatic excess pressure needed for magma-chamber rupture and ferred to as a reservoir, and as a source (ejects or injects dyke injection, and (b) the presumed depth-independence of the condi- magma) for an associated volcano. Some magma chambers, tions for magma-chamber rupture and dyke injection. particularly small-sill like chambers during their early stages The theoretical and observational considerations in the present of evolution, may be totally molten, but most magma chambers paper, however, indicate that the main differences between the pre- are partially molten, that is, porous bodies. vious models and the stress-predictions of the EWS-model rest on (3) A magma chamber that is originally with very irregular bound- some misunderstandings in the latter. In particular, for many scenar- aries is thermally (and mechanically) unstable. The host-rock ios the EWS-model assumes a lithostatic state of stress, and a cham- jogs that project into the magma tend to melt, and the ber in lithostatic equilibrium with the host rock. It also assumes magma-filled notches that project into the host rock tend to magma density equal to the host-rock density and (in some cases) solidify. It follows that the long-term equilibrium geometries zero tensile strength of the host rock. These assumptions imply that of magma chambers tend towards comparatively smooth there cannot be any extra wall-parallel stress, related to lithostatic (commonly ellipsoidal) geometries. stress, that resists magma-chamber (wall) rupture and dyke or (4) When modelling magma chamber stress fields, rupture, dyke sheet injection. The assumptions also imply that a chamber cannot injection and association eruptions, three pressure concepts contain magma with excess pressures of as much as several hundred are needed and must be distinguished. These are excess pres- megapascals. Furthermore, neither of these EWS-model predictions sure (pe), overpressure or driving pressure (po), and total pres- (extra wall-parallel stress; excess pressures reaching hundreds of sure (pt). Excess pressure is normally similar to the in-situ megapascals) is supported by direct measurements in drill-holes— tensile strength of the host rock, that is, a few megapascals. measurements that reach to crustal depths of as much as 9 km. Overpressure applies primarily to propagating sheet-like intru- While considerable progress has been made in recent decades in sions, such as dykes, and is the magmatic excess pressure plus understanding the formation, geometry, and crustal stress fields of the pressure related to the density difference between magma magma chambers, as well as the constraints on their excess pressures, and the host rock through which the dyke is propagating. comparatively little work has been done on excess-pressure changes Depending on this density difference (buoyancy effects), over- during eruptions and the possibility of pressure compartments. Pres- pressure can reach tens of megapascals. Total pressure is the sure compartments are common in many fluid reservoirs in the crust, excess pressure/overpressure plus the lithostatic pressure/ particularly in hydrocarbon reservoirs, and are being modelled, but stress at the point of observation in the dyke or magma magma-chamber pressure compartments have not received much at- chamber. tention. The present paper outlines some possible compartment sce- (5) During an eruption, the excess pressure normally decreases in narios (Figs. 15, 18) and show examples of fossil magma chambers the source chamber until it becomes close to zero, at which where such compartments are likely to have evolved over certain pe- time the dyke-fracture normally closes and the eruption riods of time (Figs. 10, 16, 17). comes to an end. On the assumption of negligible replenish- Various aspects of the chemical/petrological evolution of magma ment and no contribution from gas exsolution to the excess chambers are comparatively well understood, but there is a need for pressure during the eruption (both assumptions may be valid much more sophisticated models and a deeper understanding of the for typical basaltic eruptions of short duration), the excess physics and tectonic evolution of magma chambers. In particular, we pressure decreases exponentially until the dyke closes at its need to learn how to model the pressure changes in double magma junction with the chamber. chambers, that is, chambers that receive magma from a deeper source (6) Standard models of magma-chamber stress fields use excess (are replenished) during an on-going eruption. While some general pressure as the only loading. In the absence of unrest periods, models exist on various aspect of this problem (e.g., Woods and most magma chambers should be in a lithostatic equilibrium Huppert, 2003; Gudmundsson, 2006), more detailed work is needed. with their host rocks. Lithostatic equilibrium implies lithostatic In particular, the potential effects of compartments and double- state of stress so that all the principal stresses must be equal in magma chambers on the excess-pressure distribution in the magma magnitude and either parallel or perpendicular to the chamber, chamber need to be studied since these affect the length of (and com- that is, σ1=σ2=σ3. And these stresses must, by definition, all positional changes during) the associated volcanic eruptions. be equal to the total fluid pressure (the lithostatic pressure) in Replenishment and compartments also relate to the question of the chamber at every point at its surface. magma-chamber lifetimes. For how long, and in which way, is a (7) The standard models suggest as follows. The magmatic excess magma chamber maintained as an active source for a polygenetic vol- pressure before chamber rupture and dyke (or sheet or sill) in- cano? Many volcanoes are active for hundreds of thousands of years jection from the chamber is roughly equal to the in-situ tensile and some for millions of years. Frequent replenishment of the strength of the host rock, or between about 0.5 and 9 MPa and magma chamber is necessary so as to maintain its function as a source most commonly between 1 and 6 MPa. (b) The stress condition for hundreds of thousands of years (Huppert and Sparks, 1980; for magma-chamber rupture can be reached in two basic A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41 39

ways: (i) through increasing the total pressure inside the Acocella, V., Neri, M., 2009. Dike propagation in volcanic edifices: overview and possible development. Tectonophysics 471, 67–77. chamber (for example, by adding magma to the chamber or Amadei, B., Stephansson, O., 1997. Rock Stress and its Measurement. Chapman and through gas exsolution from its magma), and (ii) through ex- Hall, London. ternal extension, such as in rift zones, where the divergent Anderson, E.M., 1936. The dynamics of formation of cone sheets, ring dykes and caul- dron subsidences. Proceedings of the Royal Society of Edinburgh 56, 128–163. plate movements gradually reduce the minimum principal Andrew, R.E.B., Gudmundsson, A., 2008. Volcanoes as elastic inclusions: their effects on compressive stress σ3. Both loading conditions increase the the propagation of dykes, volcanic fissures, and volcanic zones in Iceland. Journal chamber excess pressure. The second type of loading (ii) gen- of Volcanology and Geothermal Research 177, 1045–1055. erally favours the injection of vertical dykes, whereas the first Annen, C., Sparks, R.S.J., 2002. Effects of repetitive emplacement of basaltic intrusions on the thermal evolution and melt generation in the crust. Earth and Planetary Sci- type of loading (i) sometimes favours vertical dykes, and ence Letters 203, 937–955. sometimes inclined sheets (and occasionally sills). Bachmann, O., Bergantz, G.W., 2004. On the origin of crystal-poor rhyolites: extracted – (8) The local stress field around a magma chamber depends, in ad- from batholitic crystal mushes. Journal of Petrology 45, 1565 1582. Balme, M.R., Rocchi, V., Jones, C., Sammonds, P.R., Meredith, P.G., Boon, S., 2004. Frac- dition to the general loading conditions, on the magma- ture toughness measurements on igneous rocks using a high-pressure, high- chamber shape, its depth below the earth's surface, and the temperature rock fracture mechanics cell. Journal of Volcanology and Geothermal mechanical properties, in particular the mechanical layering, Research 132, 159–172. Bear, J., 1972. Dynamics of Fluids in Porous Media. Elsevier, Amsterdam. of the host rock. This is demonstrated by recent analytical Bell, F.G., 2000. Engineering Properties of Soils and Rocks, 4th ed. Blackwell, Oxford. and numerical models showing the effects of layering on the Beswick, A.E., 1965. A study of the Slaufrudalur Granophyre Intrusion, South-East local stresses and surface deformation in volcanoes during Iceland. PhD Thesis, Imperial College, London. Blake, D.H., 1966. The net-veined complex of the Austurhorn intrusion, southeastern excess-pressure changes in magma chambers. Iceland. Journal of Geology 74, 891–907. (9) Some recent magma-chamber modelling takes a very diver- Blake, S., 1981. Volcanism and the dynamics of open magma chambers. Nature 289, gent view as to the loading conditions and, in particular, the ef- 783–785. Bonaccorso, A., Cianetti, S., Giunchi, C., Trasatti, E., Bonafede, M., Boschi, E., 2005. Ana- fects of gravity on the local stresses at the chamber boundary. lytical and 3D numerical modeling of Mt. Etna (Italy) volcano inflation. Geophysi- In particular, these models claim that (a) there are extra cal Journal International 163, 852–862. wall-parallel components of the lithostatic stress that act Bonafede, M., Ferrari, C., 2009. Analytical models of deformation and residual gravity – against magma-chamber rupture, (b) magmatic excess pres- changes due to a Mogi source in a viscoelastic medium. Tectonophysics 471, 4 13. Bonafede, M., Dragoni, M., Quareni, F., 1986. Displacement and stresss fields produced sures before chamber rupture can be so high as several hun- by a centre of dilation and by a pressure source in a viscoelastic half-space: appli- dred megapascals, and (c) the excess pressure that the cation to the study of ground deformation and seismic activity at Campi Flegrei, – chamber can tolerate without rupture increases with the Italy. Geophysical Journal of the Royal Astronomical Society 87, 455 485. Boresi, A.P., Sidebottom, O.M., 1985. Advanced Mechanics of Materials, 4th ed. Wiley, chamber depth. A detailed study of these models indicates New York. that some of their conclusions rest on misunderstandings as Bradley, J., 1965. Intrusion of major dolerite sills. Transactions of the Royal Society of to the implications of a lithostatic state of stress. Furthermore, New Zealand 3, 27–55. Bunger, A.P., Cruden, A.R., 2011. Modeling the growth of laccoliths and large mafic sills: the predicted stress and pressure variations are not supported role of magma body forces. Journal of Geophysical Research 116, B02203. http:// by worldwide measurements of crustal stresses and in-situ dx.doi.org/10.1029/2010JB007648. tensile strengths in drill holes, extending to crustal depths of Canales, J.P., Nedimovic, M.R., Kent, G.M., Carbotte, S.M., Detrick, R.S., 2009. Seismic re- flection images of a near-axis melt sill within the lower crust at the Juan de Fuca 9 km. None of these measurements indicate any extra wall- ridge. Nature 460, 89–93. parallel stresses nor do they show any evidence of a large Cargill, H.K., Hawkes, L., Ledeboer, J.A., 1928. The major intrusions of south-eastern Ice- depth-related increase in in-situ tensile strengths (roughly land. Quaterly Journal of the Geological Society of London 84, 505–539. Carmichael, R.S., 1989. Practical Handbook of Physical Properties of Rocks and equal to the excess pressures at rupture) which nowhere Minerals. CRC Press, Boca Raton, Boston. reach measured values in excess of about ten megapascals. Chevallier, L., Woodford, A., 1999. Morpho-tectonics and mechanism of emplacement (10) While there has been a significant progress in our understand- of the dolerite rings and sills of the western Karoo, South Africa. South African – ing of magma-chamber formation, geometry, and stress fields Journal of Geology 102, 43 54. Chilingar, G.V., Serebryakov, V.A., Robertson, J.O., 2002. Origin and Prediction of Abnor- in the past decades, as well as in understanding their chemical mal Formation Pressures. Elsevier, Amsterdam. and petrological evolution, there has been much less progress Christensen, R.M., 1982. Theory of Viscoelasticity. Dover, New York. as to the internal structural development and properties of Clemens, J.D., Mawer, C.K., 1992. Granitic magma transport by fracture propagation. Tectonophysics 204, 339–360. chambers. In particular, potential pressure compartments, Daneshy, A.A., 1978. Hydraulic fracture propagation in layered formations. AIME Society some of which may differ widely in their mechanical (and of Petroleum Engineers Journal 33–41. chemical) properties, have received little attention in Davis, P.M., 1986. Surface deformation due to inflation of an arbitrarily oriented triaxial ellipsoidal cavity in an elastic half space, with reference to Kilauea volcano, Hawaii. volcanology. Understanding such compartments requires the Journal of Geophysical Research 91, 7429–7438. combination of analysis of fossil magma chambers (plutons), De Natale, G., Pingue, F., 1993. Ground deformation in collapsed calderas structures. geophysical studies of active magma chambers, and analytical Journal of Volcanology and Geothermal Research 57, 19–38. Deming, D., 2002. Introduction to Hydrogeology. McGraw-Hill, New York. and numerical modelling as to the potential effects of compart- Dobran, F., 2001. Volcanic Processes. Kluwer, New York. ments on the transport of heat and melt between the different Domenico, P.A., Schwartz, F.W., 1998. Physical and Chemical Hydrogeology, 2nd ed. parts of the chamber prior to and during volcanic eruptions. Wiley, New York. Dzurisin, D., 2006. Volcano Deformation: New Geodetic Monitoring Techniques. Springer, Berlin. Economides, M.J., Nolte, K.G. (Eds.), 2000. Reservoir Stimulation. Wiley, New York. Acknowledgements Fjeldskaar, W., Helset, H.M., Johansen, H., Grunnaleite, I., Horstad, I., 2008. Thermal modelling of magmatic intrusions in the Gjallar Ridge, Norwegian Sea: implications I thank the Editor Lionel Wilson and the Managing Editor Timothy for vitrinite reflectance and hydrocarbon maturation. Basin Research 20, 143–159. Folch, A., Marti, J., 1998. The generation of overpressure in felsic magma chambers. Horscroft for inviting me to write this review of our knowledge of the Earth and Planetary Science Letters 163, 301–314. tectonic aspects of magma chambers with a focus on divergent views Folch, A., Fernandez, J., Rundle, J.B., Marti, J., 2000. Ground deformation in a viscoelastic as to the associated stress fields. I also thank Maurizio Bonafede and medium composed of a layer overlying a half-space: a comparison between point and extended sources. Geophysical Journal International 140, 37–50. an anonymous reviewer for very helpful comments. Francis, E.H., 1982. Magma and sediment—I. Emplacement mechanism of late Carbon- iferous tholeiite sills in northern Britain. Journal of the Geological Society of References London 139, 1–20. Frank, F., 2003. Handbook of the 1350 active volcanoes of the world. Otto Verlag, Thun, Acocella, V., 2007. Understanding caldera structure and development: and overview of Switzerland. 192 pp. (in German). analogue models compared to natural calderas. Earth-Science Reviews 85, Furman, T., Meyer, P.S., Frey, F., 1992. Evolution of Icelandic central volcanoes: evidence from 125–160. the Austurhorn intrusion, southeastern Iceland. Bulletin of Volcanology 55, 45–62. 40 A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41

Fyfe, W.S., Price, N.J., Thompson, A.B., 1978. Fluids in the Earth's Crust. Elsevier, Amsterdam. Maccaferri, F., Bonafede, M., Rivalta, E., 2011. A quantitative study of the mechanisms Geshi, N., Kusumoto, S., Gudmundsson, A., 2010. Geometric difference between non- governing dike propagation, dike arrest and sill formation. Journal of Volcanology feeders and feeder dikes. Geology 38, 195–198. and Geothermal Research 208, 39–50. Geshi, N., Kusumoto, S., Gudmundsson, A., 2012. Effects of mechanical layering of host Macdonald, K.C., 1982. Mid-ocean ridges: fine scale tectonic, volcanic and hydrothermal pro- rocks on dike growth and arrest. Journal of Volcanology and Geothermal Research cesses within the plate boundary zone. Annual Review of Earth and Planetary Sciences 223, 74–82. 10, 155–190. Geyer, A., Marti, J., 2008. The new worldwide collapse caldera database (CCDB): a tool Machado, F., 1974. The search for magmatic reservoirs. In: Civetta, L., Gasparini, P., Luongo, for studying and understanding caldera processes. Journal of Volcanology and G., Rapolla, A. (Eds.), Physical Volcanology. Elsevier, Amsterdam, pp. 255–273. Geothermal Research 175, 334–354. MacLeod, C.J., Yaouancq, G., 2000. A fossil melt lens in the Oman ophiolite: implications Geyer, A., Marti, J., 2009. Stress fields controlling the formation of nested and over- for magma chamber processes at fast spreading ridges. Earth and Planetary Science lapping calderas: implications for the understanding of caldera unrest. Journal of Letters 176, 357–373. Volcanology and Geothermal Research 181, 185–195. Marsh, B.D., 1989. Magma chambers. Annual Review of Earth and Planetary Sciences Geyer, A., Folch, A., Marti, J., 2006. Relationship between caldera collapse and magma 17, 439–474. chamber withdrawal: an experimental approach. Journal of Volcanology and Marsh, B.D., 2000. Magma chambers. In: Sigurdsson, H. (Ed.), Encyclopedia of Volcanoes. Geothermal Research 157, 375–386. Academic Press, New York, pp. 191–206. Gray, T.G.F., 1992. Handbook of Crack Opening Data. Abingdon, Cambridge. Masterlark, T., 2007. Magma intrusion and deformation predictions: sensitivities to the Gretener, P.E., 1969. On the mechanics of the intrusion of sills. Canadian Journal of Mogi assumptions. Journal of Geophysical Research 112. http://dx.doi.org/ Earth Sciences 6, 1415–1419. 10.1029/2006JB004860 (Article number B06419). Grosfils, E.B., 2007. Magma reservoir failure on the terrestrial planets: assessing the im- McKenzie, D.P., 1984. The generation and compaction of partially molten rocks. Journal portance of gravitational loading in simple elastic models. Journal of Volcanology of Petrology 25, 713–765. and Geothermal Research 166, 47–75. McTigue, D.F., 1987. Elastic stress and deformation near a finite spherical magma body: Gudmundsson, A., 1987. Formation and mechanics of magma reservoirs in Iceland. resolution of the point source paradox. Journal of Geophysical Research 92, Geophysical Journal of the Royal Astronomical Society of London 91, 27–41. 12,931–12,940. Gudmundsson, A., 1988. Effect of tensile stress concentration around magma chambers Melan, E., 1932. Point force at internal point in a semi-infinite plate. Zeitschrift für on intrusion and extrusion frequencies. Journal of Volcanology and Geothermal Re- Angewandte Mathematik und Mechanik 12, 343–346 (in German). search 35, 179–194. Menand, T., 2008. The mechanics and dynamics of sills in layered elastic rocks and their Gudmundsson, A., 1990. Emplacement of dikes, sills and crustal magma chambers at implications for the growth of laccoliths and other igneous complexes. Earth and divergent plate boundaries. Tectonophysics 176, 257–275. Planetary Science Letters 267, 93–99. Gudmundsson, A., 2002. Emplacement and arrest of sheets and dykes in central volcanoes. Menand, T., 2011. Physical controls and depth of emplacement of igneous bodies: a re- Journal of Volcanology and Geothermal Research 116, 279–298. view. Tectonophysics 500, 11–19. Gudmundsson, A., 2006. How local stresses control magma-chamber ruptures, dyke in- Menand, T., Daniels, K.A., Benghiat, P., 2010. Dyke propagation and sill formation in a jections, and eruptions in composite volcanoes. Earth-Science Reviews 79, 1–31. compressive tectonic environment. Journal of Geophysical Research 115. http:// Gudmundsson, A., 2011a. Rock Fractures in Geological Processes. Cambridge University dx.doi.org/10.1029/2009JB006791 (Article No. B08201). Press, Cambridge. 592 pp. Menand, T., de Saint-Blanquat, M., Annen, C. (Eds.), 2011. Emplacement of magma Gudmundsson, A., 2011b. Deflection of dykes into sills at discontinuities and magma- pulses and growth of magma bodies. Tectonophysics 500, 1–112. chamber formation. Tectonophysics 500, 50–64. Milne-Thompson, L.M., 1996. Theoretical hydrodynamics, 5th ed. Dover, New York. Gudmundsson, A., Brenner, S.L., 2005. On the conditions of sheet injections and erup- Mindlin, R.D., 1936. Force at a point in the interior of a semi-infinite solid. Physics 7, tions in stratovolcanoes. Bulletin of Volcanology 67, 768–782. 195–202. Haimson, B.C., Rummel, F., 1982. Hydrofracturing stress measurements in the Iceland Mogi, K., 1958. Relations between eruptions of various volcanoes and the deformations research drilling project drill hole at Reydarfjordur, Iceland. Journal of Geophysical of the ground surfaces around them. Bulletin of the Earthquake Research Institute Research 87, 6631–6649. 36, 99–134. Hansen, S.E., 1998. Mechanical properties of rocks. Report STF22 A98034. Trondheim, Moran, S.C., Newhall, C., Roman, D.C., 2011. Failed magmatic eruptions: late-stage ces- Sintef. (in Norwegian). sation of magma ascent. Bulletin of Volcanology 73, 115–122. Hansen, D.M., Cartwright, J., 2006. The three-dimensional geometry and growth of Murase, T., McBirney, A.R., 1973. Properties of some common igneous rocks and forced folds above saucer-shaped igneous sills. Journal of Structural Geology 28, their melts at high temperatures. Geological Society of America Bulletin 84, 1520–1535. 3563–3592. Hawkes, L., Hawkes, H.K., 1933. The Sandfell laccolith and “dome of elevation”. Mutter, J.C., Carbotte, S.M., Su, W.S., Xu, L.Q., Buhl, P., Detrick, R.S., Kent, G.M., Orcutt, Quarterly Journal of the Geological Society of London 89, 379–400. J.A., Harding, A.J., 1995. Seismic images of active magma systems beneath the Holmes, A., 1965. Principles of Physical Geology, 2nd ed. Nelson, London. East Pacific Rise between 17-degrees-05's and 17-degrees-35's. Science 268, Hubbert, M.K., Willis, D.G., 1957. Mechanics of hydraulic fracturing. Transactions of the 391–395. American Institute of Mining, Metallurgical and Petroleum Engineers 210, 153–168. Myrvang, A., 2001. Rock Mechanics. Norway University of Technology (NTNU), Trond- Huppert, H.E., Sparks, R.S.J., 1980. The fluid-dynamics of a basaltic magma chamber heim. (in Norwegian). replenished by influx of hot, dense ultrabasic magma. Contributions to Mineralogy O'Driscoll, B., Troll, V.R., Reavy, R.J., Turner, P., 2006. The Great Eucrite intrusion of and Petrology 75, 279–289. Ardnamurchan, Scotland: reevaluating the ring–dike concept. Geology 34, Hurwitz, D.M., Long, S.M., Grosfils, E.B., 2009. The characteristics of magma reservoir 189–192. failure beneath a volcanic edifice. Journal of Volcanology and Geothermal Research Parfitt, E.A., Wilson, L., 2008. Fundamentals of Physical Volcanology. Blackwell, Oxford. 188, 379–394. Pasquare, F., Tibaldi, A., 2007. Structure of a sheet-laccolith system revealing the inter- Jaeger, J.C., 1961. The cooling of irregularly shaped igneous bodies. American Journal of play between tectonic and magma stresses at Stardalur Volcano, Iceland. Journal of Science 259, 721–734. Volcanology and Geothermal Research 161, 131–150. Jaeger, J.C., 1964. Thermal effects of intrusions. Reviews of Geophysics 2, 443–466. Petford, N., Kerr, R.C., Lister, J.R., 1993. Dike transport in granitoid magmas. Geology 21, Jaeger, J.C., Cook, N.G.W., Zimmerman, R., 2007. Fundamentals of Rock Mechanics, 3rd 845–848. ed. Wiley, London. Petford, N., Cruden, A.R., McCaffrey, K.J.W., Vigneresse, J.L., 2000. Granite magma for- Jaupart, C., Tait, S., 1995. Dynamics of differentiation in magma reservoirs. Journal of mation, transport and emplacement in the Earth's crust. Nature 408, 669–673. Geophysical Research 100, 17,615–17,636. Petraske, K.A., Dennis, S.H., Shaw, R., 1978. Mechanics of emplacement of basic intru- Kavanagh, J.L., Menand, T., Sparks, R.S.J., 2006. An experimental investigation of sill for- sions. Tectonophysics 46, 41–63. mation and propagation in layered elastic media. Earth and Planetary Science Let- Pinel, V., Jaupart, C., 2003. Magma chamber behavior beneath a volcanic edifice. Journal ters 245, 799–813. of Geophysical Research 108 (Art. 2072, February 4). Kilburn, C.J., 2000. Lava flows and flow fields. In: Sigurdsson, H. (Ed.), Encyclopedia of Poland, M., Hamburger, M., Newman, A., 2006. The changing shapes of active volca- Volcanoes. Academic Press, New York, pp. 291–305. noes: history, evolution, and future challenges for volcano geodesy. Journal of Vol- Kilburn, C.J., McGuire, B., 2001. Italian Volcanoes. Terra Publishing, Harpenden (UK). canology and Geothermal Research 150, 1–13. Kusumoto, S., Gudmundsson, A., 2009. Magma-chamber volume changes associated Pollard, D.D., 1976. Form and stability of open hydraulic fractures in the earth's crust. with ring-fault initiation using a finite-sphere model: application to the Aira calde- Geophysical Research Letters 3, 513–516. ra, Japan. Tectonophysics 471, 58–66. Pollard, D.D., Johnson, A.M., 1973. Mechanics of growth of some laccolithic intrusions Kusumoto, S., Takemura, K., 2005. Caldera geometry determined from by the depth of in the Henry Mountains, Utah, II. Bending and failure of overburden layers and the magma chamber. Earth Planets Space 57, 17–20. sill formation. Tectonophysics 18, 311–354. Lamb, H., 1932. Hydrodynamics, 6th ed. Cambridge University Press, Cambridge. Rivalta, E., Bottiner, M., Dahm, T., 2005. Buoyancy-driven fracture ascent: experiments Lister, J.R., Kerr, R.C., 1991. Fluid-mechanical models of crack propagation and their ap- in layered gelatine. Journal of Volcanology and Geothermal Research 144, plication to magma transport in dykes. Journal of Geophysical Research 96, 273–285. 10,049–10,077. Roobol, M.J., 1974. Geology of the Vesturhorn intrusion, SE Iceland. Geological Long, S.M., Grosfils, E.B., 2009. Modeling the effect of layered volcanic material on magma Magazine 111, 273–285. reservoir failure and associated deformation, with application to Long Valley Caldera, Rosi, M., Papale, P., Lupi, L., Stoppato, M., 2003. Volcanoes: Buffalo. Firefly Books. California. Journal of Volcanology and Geothermal Research 186, 349–360. Rubin, A.M., 1995. Propagation of magma-filled cracks. Annual Review of Earth and Maaloe, S., Scheie, A., 1982. The permeability controlled accumulation of primary Planetary Sciences 23, 287–336. magma. Contributions to Mineralogy and Petrology 81, 350–357. Ryan, M.P., 1993. Neutral buoyancy and the structure of mid-ocean ridge magma Maccaferri, F., Bonafede, M., Rivalta, E., 2010. A numerical model of dyke propagation in reservoirs. Journal of Geophysical Research 98, 22,321–22,338. layered elastic media. Geophysical Journal International 180, 1107–1123. Saada, A.S., 2009. Elasticity: Theory and Applications. Krieger, Malabar, Florida. A. Gudmundsson / Journal of Volcanology and Geothermal Research 237-238 (2012) 19–41 41

Satter, A., Ghulam, Iqbal, G.M., Buchwalter, J.L., 2008. Practical Enhanced Reservoir Tada, H., Paris, P.C., Irwin, G.R., 2000. The Stress Analysis of Cracks Handbook. Del Engineering. PennWell, Tulsa. Research Corporation, Hellertown, PA. Saunders, S.J., 2001. The shallow plumbing system of Rabaul caldera: a partially intrud- Taisne, B., Tait, S., Jaupart, C., 2011. Conditions for the arrest of vertical propagating ed ring fault? Bulletin of Volcanology 63, 406–420. dyke. Bulletin of Volcanology 73, 191–204. Savin, G.N., 1961. Stress Concentration Around Holes. Pergamon, New York. Tait, S., Jaupart, C., 1989. Pressure, gas content and eruption periodicity of a shallow, Schon, J.H., 2004. Physical Properties of Rocks: Fundamentals and Principles of crystallizing magma chamber. Earth and Planetary Science Letters 92, 107–123. Petrophysics. Elsevier, Oxford. Tan, S.C., 1994. Stress Concentrations in Laminated Composites. Technomic, Basel. Schultz, R.A., 1995. Limits on strength and deformation properties of jointed basaltic Thorarinsson, S.B., Tegner, C., 2009. Magma chamber processes in central volcanic rock masses. Rock Mechanics and Rock Engineering 28, 1–15. systems of Iceland: constraints from layered gabbro of the Austurhorn intrusive Secor, D.T., Pollard, D.D., 1975. Stability of open hydraulic fractures in the earth's crust. complex. Contributions to Mineralogy and Petrology 158, 223–244. Geophysical Research Letters 2, 510–513. Thordarson, T., Larsen, G., 2007. Volcanism in Iceland in historical time: volcano types, Segall, P., 2010. Earthquake and Volcano Deformation. Princeton University Press, eruption styles and eruptive history. Journal of Geodynamics 43, 118–152. Princeton. Thordarson, T., Self, S., 1993. The Laki (Skaftar Fires) and Grimsvotn eruptions in Sibbett, B.S., 1988. Size, depth and related structures of intrusions under stratovol- 1783–1785. Bulletin of Volcanology 55, 233–263. canoes and associated geothermal systems. Earth-Science Reviews 25, 291–309. Trasatti, E., Giunchi, C., Bonafede, M., 2005. Structural and rheological constraints on Siebert, L., Simkin, T., Kimberly, P., 2010. Volcanoes of the World, 3rd ed. University of source depth and overpressure estimates at Campi Flegrei Caldera, Italy. Journal California Press, London. of Volcanology and Geothermal Research 144, 105–118. Simkin, T., Siebert, L., 2000. Earth's volcanoes and eruptions: an overview. In: Sigurdsson, Trasatti, E., Bonafede, M., Ferrari, C., Giunchi, C., Berrion, G., 2011. On deformation H. (Ed.), Encyclopaedia of Volcanoes. Academic Press, New York, pp. 249–261. sources in volcanic areas: modeling the Campi Flegrei (Italy) 1982–84 unrest. Singh, S.C., Crawford, W.C., Carton, H., Seher, T., Combier, V., Cannat, M., Canales, J.P., Earth and Planetary Science Letters 306, 175–185. Dusunur, D., Escartin, J., Miranda, J.W., 2006. Discovery of a magma chamber and Valko, P., Economides, M., 1995. Hydraulic Fracture Mechanics. Wiley, New York. faults beneath Mid-Atlantic Ridge hydrothermal field. Nature 442, 1029–1032. Wadge, G., 1981. The variation of magma discharge during basaltic eruptions. Journal Sinton, J.M., Detrick, R.S., 1992. Mid-ocean magma chambers. Journal of Geophysical of Volcanology and Geothermal Research 11, 139–168. Research 97, 197–216. Walker, B.A., Miller, C.F., Claiborne, L.L., Wooden, J.L., Miller, J.S., 2007. Geology and Sneddon, I.N., Lowengrub, M., 1969. Crack Problems in the Classical Theory of Elasticity. geochronology of the Spirit Mountain batholith, southern Nevada: implications Wiley, New York. for timescales and physical processes of batholith construction. Journal of Volca- Sparks, R.S.J., Huppert, H.E., Turner, J.S., 1984. The fluid-dynamics of evolving magma nology and Geothermal Research 167, 239–262. chambers. Philosophical Transactions of the Royal Society of London A 310, 511–534. Wang, H.F., 2000. Theory of Linear Poroelasticity. Princeton University Press, Princeton. Spence, D.A., Sharp, P.W., Turcotte, D.L., 1987. Buoyancy-driven crack-propagation—a Warpinski, N.R., 1985. Measurement of width and pressure in a propagating hydraulic mechanism for magma migration. Journal of Fluid Mechanics 174, 135–153. fracture. Society of Petroleum Engineers Journal 25, 46–54. Spera, F.J., 2000. Physical properties of magmas. In: Sigurdsson, H. (Ed.), Encyclopedia West, M., Menke, W., Tolstoy, M., Webb, S., Sohn, R., 2001. Magma storage beneath of Volcanoes. Academic Press, New York, pp. 171–190. axial volcano on the Juan de Fuca mid-ocean ridge. Nature 413, 833–836. Spera, F.J., Bohrson, W.A., 2001. Energy-constrained open-system magmatic processes Williams, J.G., 1980. Stress Analysis of Polymers, 2nd ed. Wiley, New York. I: general model and energy-constrained assimilation and fractional crystallization Wilson, L., Head, J.W., 1981. Ascent and eruption of basaltic magma on the earth and (EC-AFC) formulation. Journal of Petrology 42, 999–1018. moon. Journal of Geophysical Research 86, 2971–3001. Stasiuk, M.V., Jaupart, C., Sparks, R.S.J., 1993. On the variation of flow rate in non- Woods, A.W., Huppert, H.E., 2003. On magma chamber evolution during slow effusive explosive lava eruptions. Earth and Planetary Science Letters 114, 505–516. eruptions. Journal of Geophysical Research 108 (B8), 2403. http://dx.doi.org/ Sturkell, E., Einarsson, P., Sigmundsson, F., Geirsson, H., Ólafsson, H., Pedersen, R., De 10.1029/2002JB002019. Zeeuw-van Dalfsen, E., Linde, A.T., Sacks, I.S., Stefánsson, R., 2006. Volcano geodesy Yew, C.H., 1997. Mechanics of Hydraulic Fracturing. Gulf Publishing, Houston. and magma dynamics in Iceland. Journal of Volcanology and Geothermal Research Yu, H.-S., 2000. Cavity Expansion Methods in Geomechanics. Kluwer, London. 150, 14–34. Zang, A., Stephansson, O., 2010. Stress Fields of the Earth's Crust. Springer, Berlin. Sun, R.J., 1969. Theoretical size of hydraulically induced horizontal fractures and Zoback, M.D., 2007. Reservoir Geomechanics. Cambridge University Press, Cambridge. corresponding surface uplift in an idealized medium. Journal of Geophysical Re- search 74, 5995–6011.